I have a feeling that the private schools these elites are sending their kids to are more indoctrination centers than the government schools.
Maybe. If you go to some of those schools you're more likely to be PC etc. because of your parents, and when you come out, you're more connected to that progressive world so whatever indoctrination you get is likely to take. If you graduate from some progressive private schools, it's likely or quite possible that progressive ideology will be your future bread and butter, so you don't question it. At public schools, students can just let all that stuff (and a lot of other stuff) go in one ear and out the other.