Comparing public and private schools is very difficult. One is a monopoly, the other is competitive.
Students of competitive private schools not only pay their private school tuition, they are also forced to pay for the monopoly public school tuition of others.
Consider: if not for the "double tuition" paid by private school students, more money would obviously be available to pay teachers and other employees of provate schools. As it is now, tuition of privae schools is artificially capped by the aformentioned forced monopoly double tuition.
The forced payment of public tuition is the enemy of the private school, and the enemy of employees of private schools who desire increased compensation.
BTW you are not the only one around here who has taught in the public schools.