My expectation is that a free market of ideas would allow superior modes of education to rapidly emerge, which would then be slowly and poorly copied by the mainstream. If I'm understanding you right, you're saying that by allowing a parallel education system with freedom of choice, all the smart and rich people would abandon the old one to die. (It would be kept alive, of course by taxes, in body, if not in spirit!)
My personal belief is that the death of the old education system is over a century overdue.
We don't need to kill it, just stop making alternatives illegal and the problem will solve it itself.
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.
My personal belief is that the death of the old education system is over a century overdue.
We don't need to kill it, just stop making alternatives illegal and the problem will solve it itself.