I generally agree with your implied point: that government subsidization of education leads to price inflation in that sector, but these are private schools, where government subsidies do not (to my knowledge) apply.
The state governments do not directly fund the private institutions, but they still get a lot of government subsidies indirectly. The students can only afford to pay tuition thanks to federal loans and grants, and the professors can only afford their research thanks to government research grants.
I down voted you by accident. You are correct. Imagine a world with no 3rd party student loans. Schools would become cheaper pretty fast.