the reason you pay for private schools is specifically so you can exclude the poor

No, having a poor kid in the class does not negatively affect your kid. Having a poor performing kids that requires more resources and attention does.

No, the reason (in my case) was because the public schools prioritize social agendas over academic quality. My kids can do advanced math. My neighbor's kids estimate or guess. I hope the public schools in my area get de-funded. They are a waste of tax payer money. My choice had nothing to do with who is poor or rich.

Market rates for whom? Programmers? It was never my impression that private-school teachers were paid as well as public school benefits or had comparable benefits.

The reason you pay for private schools is the same reason that you buy a house in a good school district: to select your child's classmates and their parents.

The reason private school teachers get paid equal or less often has more to do with the level of education of the private school. The barrier to every for public school teaching in my state is very high. They get paid a pittance considering how much education they need to have to get the job and unpaid hours worked