School choice means competition among schools. Which leads to winner and losers. So you will get bad schools. The only way to avoid this is, to have only good schools, nation wide. So that the best school is the one in your district.
This doesn't follow at all. Consider e.g. a company that sells different flavors of spaghetti sauce (https://www.ted.com/talks/malcolm_gladwell_on_spaghetti_sauc...). The flavors are competing, but only on a per-person basis; overall the flavors are optimized to satisfy everyone's (different) preferences.
It's quite easy to imagine a similar concept in education, where people learn better from a particular teaching style or are better served by a particular curriculum. There's no obvious reason to expect that one "good" school would be sufficient for everyone's needs; the ideal situation would instead be that one could choose from several schools, of which at least one is "good".
I remember the case of private school in Germany, which failed to prepare their pupils for the final exam of the state and all students failed. After that it turned out their curriculum was wrong. The pupils paid the price for the failure of the school. The school closed and the pupils had to repeat the year.
School choice means competition among schools. Which leads to losers and winners. So you will get good schools. The only way to avoid this is, to have only bad schools, nation wide. So that the worst school is the one in your district.