I don't know what level of proof is necessary. I know of cases where an ISU player, Keith Sims, was granted an additional year after suddenly developing "turf toe" early in the season. Not sure how many seasons Jess Settles played -- seemed like seven or eight, probably just six.You can't just ask for a medical hardship, you have to prove it and make a strong case. Doctors complete just as much paper work than coaches or the player. The conference/NCAA will also review any medical tests or scans that were done, too, just to make sure the injury is indeed what kept them away from the field. As I mentioned before, these two injuries were pretty significant and the elbow injury probably would've been enough to get him another year, but he tried to play through it.
But there is more to it than simply "proving" an injury was serious enough to prevent the player from seeing action -- which in many cases would be hard to do. The NCAA gives the doctor/player the benefit of the doubt on that score -- as it must. However, the timing is an entirely different question.
Maybe Ott simply made a mistake by trying to come back instead of accepting the injury. Hell, the kid played through mid-season, for heaven's sake. You're arguing that he should be granted another year because he had an injury that kept him from playing -- but that clearly isn't the case, as he and his doctors thought he was still capable of playing, and he did, indeed, play.
Look at other examples. Naz Long is far from assured his petition will be granted, and his is a much more obvious case than Ott's (and falls well within the written criteria).
Virtually every football team has someone who is injured at mid-season and unable to play again. They virtually never get the year over.