Courts have ruled that government subsidies do in fact change public access. One clear example is the swarms of homeless in college libraries in inner cities. The school can't throw them out if the school took government subsidies. If the NFL owners don't want to play by taxpayers' rules, they don't get taxpayers' money.
Unless it’s a private school it’s not private property. That’s not a matter of taking subsidies, that’s a matter of schools being supported by the government for the “common good”. That’s vastly different from a private business getting a subsidy. Do you think Tesla’s factory is public property? Cause they got lots of government subsidies.