If you are perfect at setting up and managing a charity, perhaps, but then if you are concerned with education, and what something perfectly aligned with your values, may mean you need to set up a university, not a scholarship fund.

> whereas the highest paid employees of the existing "charity" are probably the president and the men's football coach

You assume that this conflicts with, rather than reflects, the interests and values of the kind of people who donate money to the universities. I think that if you explored the issue you would find that it is not the case, and particularly that the variation in whether or not that is true at any particular university correlates very neatly with the values of the people who donate to the university. (As far as causal explanations, I would assume that there is a two-way feedback loop; donors who value football will make a school more likely to pay the football coach well which will attract more donations from people who value football and less from those who do not, etc.)

> A private scholarship charity can't play those games

Yes, but that's just another way of saying that a private scholarship charity has a lot less control of what actually gets delivered as education than a university.

Not sure that's a reason to support the former over the latter, though, especially when the choice is taking the additional effort and cost to set up the former; its just another way of saying "less bang for the buck".