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.