Is that really what you think comes from public schools--a literate population? Maybe back in the 60s. Barely in the 70s. But now?
Why do you assume that parents wouldn't want their kids to be literate, if there were no public schools?
However, I am aware that the growth of literacy is tied totally to the public education movement. And I am not silly enough to believe that most people can pay the costs of private education. When it is a burden for me with an income in the upper 20% how could it be expected that a family with household income less than the median could afford a private education? Those uneducated kids become a burden on society and lower the productivity of the entire society. This is nothing something which would affect just the uneducated individuals. Just as someone running around with the plague don't just affect themselves. There are social consequences of individual choices.
When someone educates themselves s/he also helps society as a whole.