Teachers in private schools need no certification or education. That means there is a much larger pool of people to choose from. If a student is terrible, you send him to the principal, and he gets kicked out unless he is related to the principal, the teachers, or has a lot of money in a small community. You have much smaller class sizes, parents who care about their students and check their homework, and special needs students don't have to be accepted. The lower pay seems like a good trade-off to many people.
Perhaps higher salaries isn't the solution to fixing the public education system.
I agree with you on that. I'm still not going back to teaching in any districts where there are drive-by alarms or teachers are regularly physically assaulted by parents, though. It's not worth any amount of money.