SAMA should give training paths for people going into the medic fields. I always figured that with Basic, they were given training for stabilization in the field for injuries, but tougher stuff would kinda be outside of their experience levels. I feel like SAMA should be a secondary school for those in the medic fields to give additonal training for things like a heavier focus on treating field trauma, or even on ships.
Like Wes said, have SAMA oversee the VA and the Military hospitals, which could double both as places for doctors and medics from ships to retire/spend stints at, and as a place for training kind of like a residency. Medics can go to ships straight out of basic, but some time in the hospitals would be kinda looked at as finishing school to become certified for full sponsorship from the group, kind of like how a SAINT agent has to go for additional training.
SAMA can also teach courses for mental health too, both to help medics and doctors deal with mental trauma in the military and for retired vets. There are plenty of Neko still alive that more than likely have PTSD from the first and second Mishhu war, and this could fall in line with some of the stuff that Toshiro is pushing for with his character.
SAMA can also help with the research and development of new tech for medics in the field, which would help them out with things, and CDC work as well.