That's likely one of the real reasons why it isn't tried. Most of the digital modes have fairly wide signals compared to side band. A SSB signal is around 3KHz, correct, while some of the digital mode signals are like the following examples:
4K00F1E – NXDN 6.25KHz digital voice (IDAS, NEXEDGE)
4K00F1E – dPMR 6.25KHz digital voice (Typically not used in North America, used in the EU)
7K60FXE – 2-slot DMR (Motorola MOTOTRBO) TDMA voice
8K10F1E – P25 Phase 1 C4FM voice
8K30F1E – NXDN 12.5KHz digital voice (Wide DAS, NEXEDGE)
9K36F7W – Yaesu System Fusion C4FM (Voice Wide)
22K0D7E – TETRA DMO Voice
Some of those linear translators don't have that much bandwidth to begin with. Using digital isn't going to be welcomed and very highly discouraged, not because it can't be done with the right equipment.
As I said before any signal that's transmitted is always an analog type regardless of what information is being sent. The main difference is how the signal is shaped. Simple rectangular pulses are not sent because the bandwidth required is stupid wide. Various types of pulse shaping is used. One such shape is GMSK. There are others with different properties.
https://www.oscium.com/sites/default/files/WhitePaper_Simple_Signal_Shaper_GFSK.pdf
http://www.sss-mag.com/pdf/gmsk_tut.pdf
https://www.researchgate.net/profile/Thierry-Turletti/publication/2575678_GMSK_in_a_nutshell/links/0deec517e72247f39d000000/GMSK-in-a-nutshell.pdf?origin=publication_detail
I get the point about possible signal inversion occurring in a linear translator. However that would depend on the design. I would guess not all of then do so. That how the convention between using upper or lower side band on HF originated, selecting the high or low side mixing product of the VFO with the local oscillator depending on the band.
Also likely the reason why simply connecting the output from a receiver's discriminator to a transmitter's modulator might not work has to do with the frequency accuracy. With 4FSK it's critical the frequency shifts are right. Being off too much the RX radio can't decode the data. That means the level being fed into the transmitter's modulator has to be very carefully controlled to achieve the expected frequency shifts. That's also assuming the frequency shift is a linear function of the modulation voltage, which it might not be.