Nope. I don't know a single private school in the expensive bay area that pays their teachers anywhere near what public schools in the same area do. We have better retirement packages as well. As the article states, the appeal of a private school for teachers is that it is easier to teach (and typically has a religious component.) The best private schools in the area are the best because of the reasons stated in the article, not because they pay their teachers well.
I am sure universities are a different ballgame, but this is the reality of K-12 education. But hey, don't take my word for it, just my career for the last 18 years.