Then damn right there should be salary limits. Like any other public supported position.
As for Harvard argument, they are private so they can do what they want and pay what they want. When it comes to athletics, its a no brainer that the state schools (which make up the vast majority across the land) would band together to say lets make sure the private schools can't outspend us. So they create a sanctioning body that says here are the rules to compete with us. In theory, thats the NCAA. To the point we are discussing today, the NCAA has never tried to put a cap on anything. Its piles and piles of rules like you can't give a kid a t-shirt on certain days, but you can hire 50 coaches if you want. And you can give money to a charitable uhhh church (clemson). And you can write blank checks for toys. Bottom line you can outspend each other with no checks but you still have to adhere to all their other rules to ensure "fair" competition