What about government contractors? They're funded by the government. Your way of thinking about it taints every dollar that passes through government, forcing everything that deals with the government to be treated as part of the government.
You're going to have to do a better job articulating a meaningful difference between the two and why it should matter in this case. Money is money, and control is control. The government is already in the business of running schools. If it wanted to, it could open up another Harvard right down the street and fund and run that however it wants to. The government doesn't give money so that it can control Harvard, it gives money to Harvard so that Harvard can control the money.
If you want to change that, then you need to lead the charge. But it's a political question and not a constitutional one.
Private/public is, to me, not binary. There are degrees. A homeowner, for instance, does get a favored tax status where it comes to income taxes (though the home owner also pays heavy property taxes that largely offset this). Yet we wouldn't regulate what people are allowed to talk about in their own home.
But on this continuum, Harvard is pretty far toward the "public" side of private. Massive amounts of government funding and tax exemptions.
And anyway, the government wouldn't need to get in the business of running Harvard, it could just say "here are the criteria for receiving federal grants and the tax exemptions". Harvard would be free to choose.
Aren't you forgetting something? Money never comes with no strings attached. Harvard's government funding already came with conditions. Controlling Harvard's admittance policy (which has been in place for a long time) is evidently not something the government cares about.
Yes, it does actually.