Just because there’s a government subsidy doesn’t make it public property. And yes they do get to run security how they want. Even if it’s stupid.
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.