WyoJoe

I second what BoxCar said. Antenna placement is key in this situation.
The Baofeng UV-5X (aka UV-5G) is a decent GMRS radio, and, unlike the Midland (to the best of my knowledge), has a detachable antenna. For $17 with the 3800 mAh battery, that's a tough deal to beat, and it shares compatibility with many UV-5R accessories, with the exception of antennas.
If you loosen the set screw and remove the antenna, you can connect an external antenna to the Baofeng (or many of the other handheld radios).
With the right external antenna on your ATV/SxS, you should be able to get the range you're seeking, barring any major obstructions. With GMRS, the range is basically "line of sight." If there's a small hill between two units, they 'might' work, but if you're on opposite sides of a mountain, they most likely won't work to communicate between them.
Other handheld radio models to consider include the Tidradio TD-H5/Radioddity GM30/Pofung P15UV triplets (same radio, but with cosmetic differences), the Baofeng UV9G (water resistant), Retevis RT-76 or RT-76P, or several models by Wouxon.
On your SxS, you might want to consider installing a "mobile" radio rather than a handheld. The Anytone AT-779UV/Radioddity DB20G/Retevis RA25 triplets are generally regarded well on this forum by myself and others, especially at the (roughly) $100 price point. There are some other lower priced mobile units that are "water resistant," although I don't recall which models. They go up in price to about $500 for the new 50W Midland GMRS radio. Something else to consider is a used "Land Mobile" radio that is reprogrammed for GMRS. They are less expensive than many new mobile radios, and offer a good price/performance ratio.
Unless you strictly want simplicity, I'd recommend avoiding the lower-end (5 to 15W models) Midlands, as they are feature-limited, and may not suit your needs if you advance in the GMRS world.

I second what BoxCar said. Antenna placement is key in this situation.
The Baofeng UV-5X (aka UV-5G) is a decent GMRS radio, and, unlike the Midland (to the best of my knowledge), has a detachable antenna. For $17 with the 3800 mAh battery, that's a tough deal to beat, and it shares compatibility with many UV-5R accessories, with the exception of antennas.
If you loosen the set screw and remove the antenna, you can connect an external antenna to the Baofeng (or many of the other handheld radios).
With the right external antenna on your ATV/SxS, you should be able to get the range you're seeking, barring any major obstructions. With GMRS, the range is basically "line of sight." If there's a small hill between two units, they 'might' work, but if you're on opposite sides of a mountain, they most likely won't work to communicate between them.
Other handheld radio models to consider include the Tidradio TD-H5/Radioddity GM30/Pofung P15UV triplets (same radio, but with cosmetic differences), the Baofeng UV9G (water resistant), Retevis RT-76 or RT-76P, or several models by Wouxon.
On your SxS, you might want to consider installing a "mobile" radio rather than a handheld. The Anytone AT-779UV/Radioddity DB20G/Retevis RA25 triplets are generally regarded well on this forum by myself and others, especially at the (roughly) $100 price point. There are some other lower priced mobile units that are "water resistant," although I don't recall which models. They go up in price to about $500 for the new 50W Midland GMRS radio. Something else to consider is a used "Land Mobile" radio that is reprogrammed for GMRS. They are less expensive than many new mobile radios, and offer a good price/performance ratio.
Unless you strictly want simplicity, I'd recommend avoiding the lower-end (5 to 15W models) Midlands, as they are feature-limited, and may not suit your needs if you advance in the GMRS world.

The GM30 antenna can be changed, but there is a small (allen head) set screw that needs to be loosened first.
As for the belt clip, the stock one is a little flimsy, or at least it feels that way. A UV-5R or GT-3 belt clip is a good alternative, for a couple of dollars. I like and use the GT-3 belt clip personally.
One option you might consider over the GM30 is the Tidradio TD-H5. It comes in a two-pack with extra batteries and hand-held speaker mics for about twice the cost of the GM30 by itself. It is basically the same radio, just with different branding and very minor cosmetic changes. The TD-H5 works with the GM30 software, batteries, and other accessories.

When your radio is lighting up, but not putting out any sound, it is likely due to the receive tone doing what it's supposed to do. It is filtering out radio traffic that does not transmit the selected tone. That traffic could be coming from one of the repeaters you are trying to listen to, or it could be coming from some other source. It could be simplex (radio to radio) traffic that is not from a repeater. It could also be from a different repeater than you think you're listening to. If you set a receive tone to match a particular repeater, you will only hear that repeater or other traffic that is using the same tone.
I suspect what you're experiencing is the radio working normally. My suspicion is that what you thought was traffic from the repeater you programmed the radio for, was actually coming from somewhere else.

Try the following:
Tone Mode: Tone
Tone: 141.3
ToneSql: leave blank
Frequency, Offset, and Duplex all look good.
Also, leave all DTCS columns blank.
Those settings should encode your transmissions so the repeater will receive them, but also leave the code off so your radio receives everything on that frequency.

I am pretty sure they don't not have any.

I would start with nearby repeaters for your area. An easy way to get a list of repeaters is to download and use the program "Chirp." It works on Windows, Mac, and Linux, and can be used for programming a whole slew of radios.
And if your radio isn't supported? You can still use Chirp to build a file of local ham repeaters, or distant ones, or whatever you choose.
How? Open Chirp. On the menu, select Radio > Query Data Source > RepeaterBook > RepeaterBook Political Query
From there, select the state, the county, and the band, and Chirp will build a file for you of those repeaters from the query.
You can then open the file and see the location of each repeater and the configuration information for it.
If you have a compatible radio, you can even copy and paste the information into a new file to write to your radio. If you don't have a compatible radio, you would have to manually enter that information into the radio or use another software program to program the radio. Unfortunately, most of the other software applications won't allow you to copy and paste from Chirp.
 

On mine, with the computer software, I just click on the frequency field of a blank channel, and type in the frequency. I've done this to add a local ham repeater that I sometimes monitor.
The only channels where you get the "pick list" you describe is on the DIY channels up to channel 54. Try entering your new frequency on a channel higher than that.
The DIY channels allow you to program your radio for additional repeaters on the same frequencies, but with different CTCSS tones/DCS codes. That way, you can just switch channels to switch between them rather than having to edit the tone or code for the channel you're on.
Above channel 54, the frequency field is a free-form field, so you can enter any value within the receive range of the radio. These channels are set for receive only, but cover a moderate amount of the UHF/VHF spectrum.

Chirp does not work (yet) with the GM-30 or any of its clones. Fortunately, the factory software is tolerable, for me at least.

With GMRS, you'll often find that in order to have a conversation through a repeater, you have to provide the user at both ends of the conversation. With ham radio, it's very different. There is often someone tuned to the repeater who will respond when someone else keys up. Many GMRS users are people looking to communicate with their family (who are most likely using the same license). They often only use their radios when they are somehow physically separated but want to keep in touch with each other.
I'm looking into purchasing property that goes up to a ridge, and on the other side is a lake. If I put a repeater on the ridge, my wife could use the radio to keep in touch with me if I'm working somewhere on the property or kayaking on the lake. After I go back in the house, there wouldn't be much reason to leave the radio on since she could talk to me at home. That is the type of use that I've commonly seen or heard about for GMRS.

Try the following:
Tone Mode: Tone
Tone: 141.3
ToneSql: leave blank
Frequency, Offset, and Duplex all look good.
Also, leave all DTCS columns blank.
Those settings should encode your transmissions so the repeater will receive them, but also leave the code off so your radio receives everything on that frequency.

Are you planning to access any repeaters from you base station? If so, something to consider is that there could be more than one repeater on a particular frequency pair within range of you. To avoid having to change the CTCSS/DCS access code each time you switch between them, you'll want a radio where you can have multiple channels for each repeater pair you intend to use. Some radios allow this out of the box, while others (some Midlands, for example), only give you 8 repeater channels which are frequency locked to the available frequency pairs.
Used commercial radios could work well for you, but in some cases, may not offer enough channels for your purposes. Beyond that, at the upper end of the power spectrum, there is the Wouxon KG-1000G that I'm sure would offer what you need, the Midland MXT-500 (50 watt) and MXT-400 (40 watt) radios (although they may have channel limitations), and the B-Tech 50X1 (I don't recall the model number for sure, but it does have channel limitations).
In the mid-power (15-25 watt) range, there is a Wouxon model I don't recall the name of, the Radioddity DB-20G/Anytone AT-779UV/Retevis RA-25 triplets which are what I use, the Radioddity DB-25G, and a couple more Midland models, too, as well as more used commercial radios to consider. Again, some radios are limited on the number of channels and/or how you can program each of them so some of them may not work well for you.
I'm pretty happy with the DB-20G for both a mobile and base unit. It allows me to add any custom channels I need or want, including receive-only UHF/VHF channels outside the GMRS spectrum. It's not 50W, but in my case, I don't think I'd need to be.

Sounds like a keeper to me. You got a great "feel good" meter that is capable of boosting your ego as well as the watts. Perfect for the 11 meter enthusiasts.

Even experienced engineers forget at times test equipment can lie, and very convincingly too. The best test gear you own is your head, use experience and common sense.

Is the duplexer already tuned for your desired frequency? Is vendor tuning the duplexer for your desired frequency?
If no in both cases, then you'll have to get the duplexer tuned. I assume you know this already, but wanted to mention it since the duplexer in the repeater will likely be tuned for a different frequency than what you plan to use.
I really like the VXR-7000 and have found it to be about the most cost-effective repeater solution available. For mine, I went with a flat-pack duplexer that I installed inside the unit, so it's self contained. I had to fabricate a mount for it because I didn't have the factory mount, and I don't think it would have worked with the duplexer I bought.
I just made two "straps" out of 1/8" x 3/4" flat stock, and bought metric screws to attach to the factory mounting holes. I used a little foam padding for cushion and strapped the duplexer to the repeater, and arranged the cables as necessary.
It is working well for me.
I also bought a programming cable with the CE-27 program disk which allows me to program the VXR-7000 easily.

In spite of the occasional programming issues, I really like my AT-779UV / DB-20G radios. From a price/performance perspective, I think this is where you can get the best bang for the buck in a new radio. There may be used commercial radios available at around the same price that offer better performance, but they won't be new at that price, and they have to be programmed, requiring software and a programming cable. Depending on the radio, that can quickly add a lot to the price.

I've used a hand held radio in a mobile configuration simply by connecting a mag mount antenna on top of the car and using a battery eliminator to power it.
Most of the time, I'm just monitoring the radio, but with only those improvements, the reception is much better than with the rubber ducky antenna.
I can't speak much to transmitting, though, because I haven't tried using it outside of the normal range where the rubber ducky would work.

I've used a hand held radio in a mobile configuration simply by connecting a mag mount antenna on top of the car and using a battery eliminator to power it.
Most of the time, I'm just monitoring the radio, but with only those improvements, the reception is much better than with the rubber ducky antenna.
I can't speak much to transmitting, though, because I haven't tried using it outside of the normal range where the rubber ducky would work.

As @OffRoaderX mentioned, the GMRS repeater channels share the same output frequency as the "simplex" channels 15-22. Here's a long-winded explanation:
Just to clarify, a repeater is a system that takes the output from your radio and re-transmits it, generally with more power and from a better location. This allows you, through the repeater, to get a signal out to a wider area, and potentially, to a wider audience. The repeater channels (numbered 23-30 on many radios) receive the signal from your radio in the 467 MHz range, and re-transmits the signal in the 462 MHz range. The exact frequency depends on the channel you select, but if you use, for example, Channel 26 (Often called "Repeater 18" or "Repeater 4" or something similar), the repeater receives at 467.625 and transmits on 462.625. Some radios call this "Repeater 18" because it uses the same transmit frequency as Channel 18. Others call it "Repeater 4" because it's the fourth "frequency pair" of the designated pairs. There are 8 in all.
Your radio, on the other hand, when tuned to that channel receives on 462.625 and transmits on 467.625 (opposite of the repeater). This allows your radio to hear the repeater and vice-versa.
Simplex communication uses the same frequency for both transmit and receive. On most radios, Channels 1-22 are simplex channels, and use the same frequencies as Channels 1-22 on FRS radios. This mode is for direct radio to radio communication without using a repeater.
Now, regarding your question about the "Emergency" channel repeater 20...
GMRS does not have a designated "Emergency" channel for the most part. There may be some communities where a channel is monitored in the event of an emergency, and there may be repeaters that are designated for use by emergency services, but that would be established within a small area, and is not common to the GMRS community as a whole.
It seems as though in your area, you may have a repeater designated for use in emergencies, that is tuned to the "Repeater 20" channel (The repeater is tuned to receive on 467.675 and transmit on 462.675). You mention a "tone" of 141.3. This is what is referred to as "continuous tone coded squelch system" or "CTCSS." This is a sub-audible tone that is generated by the transmitter, which "opens" the squelch of the receiver, allowing the message to be heard. Without the proper tone, the receiver ignores incoming transmissions, and only listens to those using the proper tone. If no receive tone is designated, the receiver will listen to all incoming signals.
If you set your radio to use tone 141.3 on transmit only, it will generate the tone needed to open the squelch of the repeater, but will receive all incoming signals regardless of the tone associated with those signals. In general, this is the configuration I use for repeater channels as it allows me to hear all incoming traffic on the frequency regardless of tone. That means I may be hearing simplex transmissions in addition to those coming from a repeater.
If you set your radio to use tone 141.3 on receive only, it will not generate the tone needed to open the squelch on the repeater, so the repeater will ignore your transmission. You will also only receive incoming signals that are encoded with that tone, and you radio will ignore all other incoming signals.
If you set your radio to use tone 141.3 on transmit and receive, it will generate the tone needed to open the squelch on the repeater, but will only receive incoming signals that are also encoded with the same tone. If you have a lot of incoming signals that are not from the repeater, this is a way to "filter" those signals out so you only hear the repeater.
There are also some repeaters that use "split tones." This is where the transmit and receive do not use the same tone. This is helpful to reduce unauthorized use of a repeater as it makes scanning for tones a bit more difficult (but not impossible). This also only affects you if you enable CTCSS on receive on your radio. I just wanted to mention it so you're aware that some repeaters use this configuration.

Even though you say this is off topic and is not about antennas, I would argue that it actually is about antennas, but not "HT" antennas.
I think the biggest difference when using an amplifier like you describe is that it uses a more sophisticated antenna system. It could be as simple as a mag-mount antenna on a pizza pan, or as complex as a tower-mounted antenna system, but either way, it's a vast improvement over the rubber ducky.
Even without the amplifier, if you set up an external antenna like what's needed with the amplifier, you will see an improvement in range.
The problem with that is that you lose the portability of the radio, and it's no longer hand held at that point.

As @OffRoaderX mentioned, the GMRS repeater channels share the same output frequency as the "simplex" channels 15-22. Here's a long-winded explanation:
Just to clarify, a repeater is a system that takes the output from your radio and re-transmits it, generally with more power and from a better location. This allows you, through the repeater, to get a signal out to a wider area, and potentially, to a wider audience. The repeater channels (numbered 23-30 on many radios) receive the signal from your radio in the 467 MHz range, and re-transmits the signal in the 462 MHz range. The exact frequency depends on the channel you select, but if you use, for example, Channel 26 (Often called "Repeater 18" or "Repeater 4" or something similar), the repeater receives at 467.625 and transmits on 462.625. Some radios call this "Repeater 18" because it uses the same transmit frequency as Channel 18. Others call it "Repeater 4" because it's the fourth "frequency pair" of the designated pairs. There are 8 in all.
Your radio, on the other hand, when tuned to that channel receives on 462.625 and transmits on 467.625 (opposite of the repeater). This allows your radio to hear the repeater and vice-versa.
Simplex communication uses the same frequency for both transmit and receive. On most radios, Channels 1-22 are simplex channels, and use the same frequencies as Channels 1-22 on FRS radios. This mode is for direct radio to radio communication without using a repeater.
Now, regarding your question about the "Emergency" channel repeater 20...
GMRS does not have a designated "Emergency" channel for the most part. There may be some communities where a channel is monitored in the event of an emergency, and there may be repeaters that are designated for use by emergency services, but that would be established within a small area, and is not common to the GMRS community as a whole.
It seems as though in your area, you may have a repeater designated for use in emergencies, that is tuned to the "Repeater 20" channel (The repeater is tuned to receive on 467.675 and transmit on 462.675). You mention a "tone" of 141.3. This is what is referred to as "continuous tone coded squelch system" or "CTCSS." This is a sub-audible tone that is generated by the transmitter, which "opens" the squelch of the receiver, allowing the message to be heard. Without the proper tone, the receiver ignores incoming transmissions, and only listens to those using the proper tone. If no receive tone is designated, the receiver will listen to all incoming signals.
If you set your radio to use tone 141.3 on transmit only, it will generate the tone needed to open the squelch of the repeater, but will receive all incoming signals regardless of the tone associated with those signals. In general, this is the configuration I use for repeater channels as it allows me to hear all incoming traffic on the frequency regardless of tone. That means I may be hearing simplex transmissions in addition to those coming from a repeater.
If you set your radio to use tone 141.3 on receive only, it will not generate the tone needed to open the squelch on the repeater, so the repeater will ignore your transmission. You will also only receive incoming signals that are encoded with that tone, and you radio will ignore all other incoming signals.
If you set your radio to use tone 141.3 on transmit and receive, it will generate the tone needed to open the squelch on the repeater, but will only receive incoming signals that are also encoded with the same tone. If you have a lot of incoming signals that are not from the repeater, this is a way to "filter" those signals out so you only hear the repeater.
There are also some repeaters that use "split tones." This is where the transmit and receive do not use the same tone. This is helpful to reduce unauthorized use of a repeater as it makes scanning for tones a bit more difficult (but not impossible). This also only affects you if you enable CTCSS on receive on your radio. I just wanted to mention it so you're aware that some repeaters use this configuration.

As @OffRoaderX mentioned, the GMRS repeater channels share the same output frequency as the "simplex" channels 15-22. Here's a long-winded explanation:
Just to clarify, a repeater is a system that takes the output from your radio and re-transmits it, generally with more power and from a better location. This allows you, through the repeater, to get a signal out to a wider area, and potentially, to a wider audience. The repeater channels (numbered 23-30 on many radios) receive the signal from your radio in the 467 MHz range, and re-transmits the signal in the 462 MHz range. The exact frequency depends on the channel you select, but if you use, for example, Channel 26 (Often called "Repeater 18" or "Repeater 4" or something similar), the repeater receives at 467.625 and transmits on 462.625. Some radios call this "Repeater 18" because it uses the same transmit frequency as Channel 18. Others call it "Repeater 4" because it's the fourth "frequency pair" of the designated pairs. There are 8 in all.
Your radio, on the other hand, when tuned to that channel receives on 462.625 and transmits on 467.625 (opposite of the repeater). This allows your radio to hear the repeater and vice-versa.
Simplex communication uses the same frequency for both transmit and receive. On most radios, Channels 1-22 are simplex channels, and use the same frequencies as Channels 1-22 on FRS radios. This mode is for direct radio to radio communication without using a repeater.
Now, regarding your question about the "Emergency" channel repeater 20...
GMRS does not have a designated "Emergency" channel for the most part. There may be some communities where a channel is monitored in the event of an emergency, and there may be repeaters that are designated for use by emergency services, but that would be established within a small area, and is not common to the GMRS community as a whole.
It seems as though in your area, you may have a repeater designated for use in emergencies, that is tuned to the "Repeater 20" channel (The repeater is tuned to receive on 467.675 and transmit on 462.675). You mention a "tone" of 141.3. This is what is referred to as "continuous tone coded squelch system" or "CTCSS." This is a sub-audible tone that is generated by the transmitter, which "opens" the squelch of the receiver, allowing the message to be heard. Without the proper tone, the receiver ignores incoming transmissions, and only listens to those using the proper tone. If no receive tone is designated, the receiver will listen to all incoming signals.
If you set your radio to use tone 141.3 on transmit only, it will generate the tone needed to open the squelch of the repeater, but will receive all incoming signals regardless of the tone associated with those signals. In general, this is the configuration I use for repeater channels as it allows me to hear all incoming traffic on the frequency regardless of tone. That means I may be hearing simplex transmissions in addition to those coming from a repeater.
If you set your radio to use tone 141.3 on receive only, it will not generate the tone needed to open the squelch on the repeater, so the repeater will ignore your transmission. You will also only receive incoming signals that are encoded with that tone, and you radio will ignore all other incoming signals.
If you set your radio to use tone 141.3 on transmit and receive, it will generate the tone needed to open the squelch on the repeater, but will only receive incoming signals that are also encoded with the same tone. If you have a lot of incoming signals that are not from the repeater, this is a way to "filter" those signals out so you only hear the repeater.
There are also some repeaters that use "split tones." This is where the transmit and receive do not use the same tone. This is helpful to reduce unauthorized use of a repeater as it makes scanning for tones a bit more difficult (but not impossible). This also only affects you if you enable CTCSS on receive on your radio. I just wanted to mention it so you're aware that some repeaters use this configuration.

One other thing I neglected to mention in my post above, if the antenna is tuned to 465 MHz, it should perform very well on 462 MHz, and should not need to be retuned.
There is more to antenna performance than just a great SWR, as I'm sure many on this forum will tell you if you ask.
Have you checked the antenna's SWR? Is it below 1:2?
People generally want a "perfect" 1:1 SWR, when in reality, an SWR below 1:2, or even higher in some cases, is acceptable.

I'm not sure what you're using for a repeater or mobile radio, but are you using the mobile in close proximity to the repeater?
If so, you may be experiencing desensing of the repeater or the radio. How are you checking to determine if the audio is coming through? Are you listening on another radio while transmitting? If so, it's probably that radio that is experiencing desense.
Since you are hearing the repeater tail, this is what I most suspect in your case.

I'm not sure what you're using for a repeater or mobile radio, but are you using the mobile in close proximity to the repeater?
If so, you may be experiencing desensing of the repeater or the radio. How are you checking to determine if the audio is coming through? Are you listening on another radio while transmitting? If so, it's probably that radio that is experiencing desense.
Since you are hearing the repeater tail, this is what I most suspect in your case.