“they get way more then teachers in private schools who have much higher demands”
Nonsense. In private schools, disruptive students can be kicked out. Not so in public schools. Public school class sizes are also usually much higher.
Private school *principals*, however, often make substantially more than public school principals even the schools are much, much smaller. I know of one principal at a Jewish Day School who was making over 400K/year, and numerous others in the 200K-300K range. The top of the salary scale for NYC public school principals is just over $150K.