A bureaucracy is run for the benefit of its own bureaucrats. This is the problem with a “public” school system. As Highland Park, MI lost population, it did not cut back on the number of education employees. Bureaucracies do not cut themselves.
A private system is more responsive to the taxpayer/consumer, because the consumer has leverage. In any private business relationship, the consumer can walk if he is not receiving the desired service, in this case effective education.
When the system is public, there is no real recourse. The consumer is stuck with the system. Sure, you can vote in a different set of bureaucrats, but they ultimately will act in their own best interests.
A private system will lead to better results involved for student and even teachers but at the expense of the bureaucrats. That is why you will continue to see union leaders and boards of ed fighting reform tooth and nail.