I don’t think 3 is true at all. I think most teachers are happy with public school indoctrination. Hell, they are the ones doing the liberal, global warming, touchy feely, zero-tolerance, anti-religion, anti-history, pro-feminist, pro-gay indoctrination. But your #1 and #2 are dead on.
I agree that the few conservative teachers fit your #3 in spades, but I’m not so sure most of those sending their children to private schools are conservatives. I think it is about education, discipline and very much about safety.
It is much less likely for your kid to suffer a beat down in private school, because the guilty kid will be summarily expelled. Kids don’t get away with sassing and threatening teachers in private schools the way they do in public schools. You don’t have the playground crime or the massive drug use in the higher grades. All that stuff is policed and violators are suspended or expelled. It is a bit of different world in private school.
All this, the public school teachers well know.
Both liberals and conservatives send their own children to private schools. The difference is that the conservative parents are proud of it, while the liberal parents feel ashamed about it.