The client is the parent, not the child. Parents tend to fall all over themselves trying to get their kids into good schools which is why property prices in highly rated school districts are always higher. Your only option is to move there, which essentially is how people of lower incomes actually do get trapped where they are...they can't afford to just up and move because the school was terrible.
Now, here's the thing: to end up with a highly rated school you need good teachers. Good teachers tend to gravitate towards the same factors that any other employee gravitates towards: pay rate and a good work environment. For a teacher, a good work environment usually means administrators who support them at all times - including dealing with parents.
Now this is a hugely important scenario to take into account. If a teacher is pressured by parents and fired by a public school currently...where exactly are they supposed to go? In a school choice scenario, they'd have multiple different options instead of many schools from one central authority. When teachers have more options and can simply walk out the door of a bad work environment with an unsupportive admin, they'll end up getting RECRUITED by other school administrators who know better.
Parents will continue beating down the door of those good schools to get in, but right now if a parent threatens a teacher, mouths off or pressures them for better grades they are doing so with no potential consequence. If you try that at a private school you know what happens? Parent gets a warning and if it happens again the child will have to find another school the next year. The admin isn't going to risk losing a great teacher to appease little Johnny's parents.
This shifts the ENTIRE dynamic of the school system to one in which teachers are in control because teachers are effectively the product.
Just think about what the job market looks like for programmers right now in areas with high demand. That's essentially what the market for teachers would become.