Lots of private schools pay worse than public schools, true, and do indeed draw teachers who are decent-to-good but sick of putting up with public school admin and parent bullshit—for reasons that are obvious if you think about it, but perhaps not intuitively obvious, private schools are less beholden to any one parent than public schools are, and are much better positioned to firmly tell them "no"—but once you start to get into actually-good private school territory[0] pay and appreciation (and to a some extent social status, at least within a certain circle) match or beat area public schools, typically.
[0] A very high percentage of private schools are not good, though—"how can that be? How do they keep getting enough students to stay open, in a competitive market that includes 'free' as an option?"—politics and religion, is the answer, and they can under-perform on academics year after year as long as they're delivering the message the parents want on those fronts. Incidentally, the teachers these schools draw with their lower pay aren't always leaving public education over legitimate bullshit, but often over perceived bullshit like "they wouldn't let me talk about Jesus as much as I'd like", and these are usually not good ones that the public schools are losing to private schools, but ones who we'd want to leave public education.