At some point, as I had a 10-11 player roster and had 1-2 injuries to my annoyance, this system is something that I tried to use to train 6 players. I also tried to have a 3-man roster for a scrimmage with one player only being a backup to get some minutes but not too many, which failed, because the GE preferred him on an unassigned position instead of a lucky fan. So, been there done that
I also have had trainees (18-19 years old) actually going to and staying at GS 9 with only 48 weekly minutes played with a lvl 5 psychologist. I have also read about how the psychologist does not have the effect one might hope for, but I tried it and it seemed to work fine (maybe I just got lucky). It is also feasible to get a lvl 5 psychologist for <6-7k salary for a relatively low price and then hold onto until the next draft. I really do not know if a player with 96 weekly minutes could maintain a good GS with the psychologist, so I cannot give you any insight there.
In any case you can let me know how it works out for you, should you try to use it. And I'm crossing fingers that your players remain injury-free, otherwise it would really become a nightmare.