That doesn't mean, though, that any level of US government has to operate and staff the schools. The Supreme Court ruling can just as easily be met through the private sector. But the catch would be that the government would have to pay for it (meeting the 'provide' mandate).
In any case, having the private sector run schools as a 'for-profit' industry would increase the quality and decrease the cost... providing that government didn't turn it into another monopoly like how some do with cable television.
They will and the government/teachers union will ensure it.