College football is a mess, with Ohio State and The U providing the latest "scandals" and with the pattern of conference jumping we've seen lately.
I think it's time to split big time football from academics. Dissolve the NCAA. Pay the players. Don't even force them to be students if they don't want to be students. Treat college football like an age 21 and under pro league. The schools rent out their facilities, names, supporters, etc. and the football program is separate from the school itself, just like the food service program.
I've long viewed college football, and college athletics in general, as a sort of "loss leader" for the university. A prominent sports program raises the university's status. Saying that it doesn't do much for the university's core mission only makes sense if you think the core mission is efficiently allocate resources towards the best educational environment possible.
But it clearly isn't. Universities exist for a host of reasons, most related to status and social networking rather than actual education. Which is why so many people are willing to pay huge premia to go to 4-year universities for basic classes rather than 2-year colleges, even though the class quality will usually be comparable or even better at the 2-year schools (b/c of smaller class sizes, professional teachers rather than researchers teaching those classes, same texts and curricula, etc.). And why many people are willing to pay even higher premia to go to flagship 4-year colleges rather than Eastern Small Town State, even though the actual education will be very similar.
So having high-profile sports programs does serve universities' core mission: it raises the university's status, and that attracts students and other sources of funding.