Nah, both public and private universities in the US are pushing this political and ideological agenda. I'm not aware of any systematic pay difference for academics in public vs private universities - for example, Stanford (private) pays similarly to Berkeley (public), and they have similar politics. Both pay much less than private sector for anyone who can get an academic job at one of those institutions.

Your comment is a little more applicable to museums and the arts, but the US government funds all universities, both public and private, pretty lavishly. Unfortunately the university administrators have a habit of accruing much of that funding to themselves and leaving the academics relatively little.