Letting the people with wealth and power use an alternative system would be the same thing as killing the public system. No one with the ability to improve the system would have the incentive to. It's the reason public schools aren't improving. In a privatized system you'll get a divided population with some being very well educated and while some get no education at all.

There's nothing preventing a public system from being changed and improved. Take Finland as an example. Everyone goes to the same public schools there, rich or poor. The result is one of the best education systems in the world. New methods are regularly tested and teachers have a lot of freedom to choose the ones that suit them.