There is no choice here because it fails the 80/20 rule of any hiring system. Any less than 20% capacity spare and there is no free choice.
So since there is no actual choice, why give people one? Is it the same as with toddlers - You can have red juice or blue juice - a way of giving the feeling of control without actually having any.
Giving people a choice and then not giving them what they want causes loss aversion responses. That is worse than not having any choice at all.
Kids want to go to the nearest school and the schools should be sufficiently equipped to cater for all needs in the area.
Every other solution I've seen is more difficult than that one and is based around psychological manipulation rather than solving the problem.
And no the 'free market' won't provide. You're supply side limited by the number of people capable and willing to be teachers. That's why you have to have a mass production system for education.