Often this is addressed through a voucher system. Single Payer (or subsidizer), multiple provider.
Eg: gov't gives a voucher for $10k per head, you're free to shop for the best schooling your money can buy (perhaps even online!)
Aside: this exact concept can apply to healthcare too
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.
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.