Falma wouldn't even have time to assume control of the place. He merely would need to be accepted as an authority and source of information, without needing to go through the slow process of convincing every single old beard individually of the merits of his knowledge. So, if the empress, Falma's rather famous father, and the church leader all vouch for him, but some stubborn bastard still refuses to even attend a single lecture demonstrating what Falma would have to offer, then it's time for such a person to retire.
Without some force behind him, despite his fame, which most definitely has reached the medical school, there indeed should be old professors who feel threatened by the possibility of losing their apex position in the medical field. But if they haven't got a choice, they would need to hear what Falma has got to say and check out his textbook. Then, perhaps, the long lost ambition might be rekindled in the heart of an old fool and they might even embrace the world of new medicine. Forced out of their comfort zone of many years and all that.
It's not like I wouldn't have experience of academics. Though my experience also says too much of a professor's valuable time is spent on worrying about financies. If you listen to Falma, there might be money, if you don't listen to him, who knows.