Public schools came into existence because educated citizens were necessary for the good of the state and the public alone was able to step in to the breach to provide them. Today however there are a surplus of educated people and private capital able to operate schools. Public schools actually work against quality education in America for the masses by displacing or aborting private school conceptions.
The public should have a role in funding private schools that could appear like private contractors providing services and materials to the Department of Defense. Quality education products and results would be rewarded and competition to flourish. People with a PhD in history could stop working at a copy shop and take a job for a while in a private school instructing high school students.
Public schools have reinforced gross and pervasive income inequality based school district financing. To receive federal support private schools funded by the public should receive the same per student funding across the state and all school districts. Students from ghetto neighborhoods would have the same per student investment as those from rich districts. Private schools not funded by the state could of course choose to opt out and not receive public funding.
Private prisons have a profit motive to incarcerate people. Youth have a per-head value of maybe $40,000 each as prisoners to the for-profit prison keeper. In fact the value to a capitalist of the profit he can make by employing an individual may be less than what he can make by incarcerating an individual. It would be better if the profit motive was applied to public education and graduating quality students rather than to incarcerating youth.