anecdotally i went through almost all of my k-12 in public school and it seems like the opposite: that areas with more expensive houses pay more in local taxes which fund nicer schools and governments. meaning the "quality" of the local government is a function of the wealth held by its residents. this is all to say that be it the local government or a privatized system, the education system seems very much beholden to their patrons, the rich families. i don't see why privatization would be a better alternative.
> I'm not sure that the government management of schools aligns the school's interests with long term goals.
saying that the current system isn't good enough doesn't automatically mean that some other system is better. the question remains.