The Free Zone Board

[Spallenzani]

can not think of a single example of a society that educated, or attempted to educate everyone, where there were strictly private schools.

Any Orthodox Jewish community is a perfect example of such a society. Every child is educated in a private school, and tuition for poor children is paid for through private donations. I myself was able to go to a Jewish private school through such philanthropy.

Essentially if one were to privatize the education of everyone, someone would still have to pay for the education of those who could not afford the tuition.

Correct. So even if you believe that private charity is not enough to pay for children's education, that is not by itself enough of a reason to support public schools. We could simply use taxes to pay for need-based scholarships instead of public schools.