I have an honest question that I don't expect anyone to honestly answer, but I'll ask it anyway.  Do you believe that public school teachers are better teachers than private school teachers?  Because if we are going by teacher salaries, which is ostensibly the measure of a teacher's value, your own argument would seem to indicate that teachers at private schools are less valuable than teachers in public schools.  

I noticed that you've not advocated increasing pay to private school teachers to bring them up to parity with public school teachers, let alone increasing their pay beyond that.  Why is that?  Are they not worth it?  Do they not deserve a pay increase?

When you frame your argument in moralistic terms, you open the door to the same kind of argument.  By my reckoning, we should be trying to increase teacher pay at private schools to at least bring it up to parity with public school teachers.  If you need a pay raise, then they most certainly do since they earn less than you.