Given the sheer amount of taxpayer dollars they absorb via tax breaks etc they should be subject to oversight. Harvard costs taxpayers billions of dollars, perhaps with good ROI given their stature but at a huge cost.

Well, the problem is that they serve as a credential issuing authority, but they also provide what is essentially a credential laundering service for the people who can pay.

I do not know specifically about Harvard, but generally when you receive a lot of federal money, you have to comply by certain rules. I don't know if there are existing rules that govern this, but if not, I'm sure plenty are trying to create those rules.

For quality of education and quality of reputation.