The obvious connection is who pays makes the rules. The US federal government uses access to Pell grants to motivate private educational institutions to do what

The obvious connection is who pays makes the rules. The US federal government uses access to Pell grants to motivate private educational institutions to do what they want, it can use access to any other funds to get Harvard to do what it wants, if it wants to. Obviously it won’t, because if banning ROTC from your university doesn’t get your access to federal funds cut off affirmative action certainly won’t but they could.