If the schools were public, then the public (at large) would control them. This is clearly not the case with the schools we have now. They are government funded, with government mandated curriculum. They are run by quasi-governmental entities elected in the same fashion as government itself.
They are government schools.
As for what to do about it (at least here in Texas) see my earlier rant.
Postscript
What an asshole. A distinction without any real difference. Government can educate children just fine, thank you. Better than private industry can, since private industry isn’t tasked with educating people as a baseline for its survival. That is a job that has been left to the government to do.
