IMHO, the whole mindset about college is broken. There was a time in this country when the value of educating someone was viewed more from the perspective of society. We wanted people to go to college to have a more informed electorate and a more professional workforce. This started to change under Ronald Reagan's governorship in California and was completed under his presidency. And that was a shift towards viewing education as a benefit to the student.
It seems a subtle distinction, but it has a crucial effect on tuition rates. When it's looked at from the perspective of the benefit to the student, the upper bounds for how much an educational institution can charge is the financial benefit over the course of that student's career. When you look at it from the standpoint of benefiting society, it's in the best interest of the state to educate students as cheaply as possible. And we've seen the predictable results of the change in perspective with tuitions slowly rising towards that upper bound.
But you're right...placing the primary funding burden on an entity that is sensitive to price will likely put pressure on colleges to offer educations more efficiently. Though I do think your focus on football coaches is unwarranted. That's a simple equation...football brings in $Xm/yr so they give it a budget of $Ym/yr. As long as the delta between X and Y is as big as possible, it doesn't matter what Y is. The far more concerning thing, to me, is how long we've known about the concussion issue at this point and how little college football programs have done about this. We can look at the time period since CTE was first discovered as willfully endangering college football players and colleges will be on the hook for billions of dollars in lawsuits over the next 2 decades. This isn't particularly relevant to the issue we're talking about, but will have major ramifications in bottom lines for Universities, so will probably impact tuitions as well.