Private entities are immune from corruption and never collude with politicians? People do not get fired for political reasons at private institutions?
You show a mechanism through which the government can abuse its power, which is terrible and everyone should agree that it’s bad and change how this works. But how not having that implies that a chancellor picked by a board with no oversight could not be worse? The problem is the lack of checks against abuse of power and mechanisms to guarantee a basic independence level for the professors. Once you have those in place, how is a university better merely by being private?
I wish free-market neo-liberals would apply to themselves what they say every time someone mentions socialism: pure ideology might sound good but does not work in the real world.