The original purpose of state universities when they were founded was to provide the public with access to a low cost, quality public education. Prior to the establishment of the state university systems, the country had all private universities that were typically attended by wealthy people. There was very little social mobility. The state universities were an attempt to remedy that.
I disagree with your views of the purpose of a state university. The purpose of a state university is not to be a bastion of progressive thought- however, that's what they have turned into through leftist professors. The state university is supposed to be a bastion of free thought, which is not the same as progressive thought.
Ultimately that's not really pertinent to my point which is that the federal government has a duty to the taxpayers to ensure that federal funds are being spent wisely. I don't think that coaching salaries should be capped simply because they are making too much money. However, if the fact that universities are paying coaching staffs millions of dollars is causing tuitions to be higher than they would otherwise be and schools are not controlling their own costs, the federal government has every right to set conditions on the universities to receive federal taxpayer money. Again, state university systems can freely decide to forgo the federal funds and pass the costs onto the students or the state taxpayer. It's like the recruiting example I gave above. Many state universities openly embraced anti-military neo-marxist students and professors that would harrass military recruiters, right up until the federal government set conditions and told universities that they HAD to provide access or they would lose funds. And wouldn't you know it, state schools suddenly became much more military friendly.
Lastly- my support of the cap only goes for public non profit universities. I don't believe it should apply to private non profits. Since private universities are not taxpayer funded, the federal government does not have an interest in their tuition costs or the amount being paid to coaches. The biggest winners will be Miami, USC, and Notre Dame.