Harvard is subsidized by the govt (in the form of low interest loans to students) and via grant money for research.
re, the first part: So you're saying that if I run some private business, and the gov't gives a loan to somebody else to buy the product I'm selling, that makes me a publicly funded business and forces me to abide by rules for businesses that are publicly funded? I don't think that makes sense.
Government financing / backing / regulation of student loans is much more complex and closely bound to colleges than you are making them out to be.
If we posted stipulations on all the grant money we give universities, every elite university would quickly follow through with it. It's too much money to throw away, even for Harvard.