Does that fall on the head coach? My guess would be that they would report separately to the GM, but it is only a guess. It's probably also a slightly less urgent change than any you might make to the coaching staff, as they're presumably not part of any FA or draft decisions.
I don't know the exact process but a treatment plan and such is created for a player and relayed. As far as I know, the coach GM shouldn't have too much input as they're not medical people but we all know that's probably not the case. Their jobs, in many cases, hinge on players not being out too long. So in the past coaches and GM's don't exactly have the players best interest at heart. I think that's changed somewhat as players now have more input then ever in their treatments.
Anyway, one year of injuries is a small sample size and not any indication of a poor medical, strength and conditioning staff, especially in Football.
What gives me pause is the handling of Osemele which was flat out weird and Moseley groin. The Jets with Pearlman and his interesting (criminal really) take on concussions and thru this year don't have a great reputation for medical staffing. So yeah I would love them to take a look at things. I don't know what that entails and how long it should take. It's the offseason so take your time. NYC has some of the finest Dr's in the world and they're not going anywhere.
The only part on Gase that I know is they put rehabbing players fines out in the open on an electronic board like a bunch of putzes, that's on the coach. Other than that the practices are governed by the CBA so I don't think the injuries are the result of anything Gase has done. Even when Williamson was injured in preseason, he begged to stay in the game. That's bad luck...
I think Gase is what he is, not very good, but the injuries are not something to pin on him. It's the NFL.