Well keep in mind that all of the characters applying will be new to the college. And again, if you intend to apply a character who is already inexplicably capable of magic before even arriving to the college, then there's no reason the college would need to accept such a person since they obviously managed just fine without them. As for obtaining prominence, that itself isn't going to be as easy as you may think either. 

While the plot will mostly be about the party, that doesn't mean the world doesn't also move at it's own tempo, and there's plenty of other people in the college competing for prestige and spots to learn Esoteric magics as well. After all, even with all the dangers there are to them there's people who are more than willing to risk it for the power such magics hold. To that end, there will be other NPC's who are doing things like trying to win competitions, do research and invent something, or otherwise also make the dean's list. Keeps PC's on their toes.