I will disagree with you that most people choose to live where they do because of school choice. While that may be in the more urban areas, in a lot of the state, people live where they live because that's where they grew up and their family is from.
Again, just my opinion here, the problem becomes when the school pays tuition for that student based on athletic prowess and can also reject having other students going there based on other reasons(not saying they do, but they can). In a public school we can't pay for a student's tuition or pay for a student's room and board. Do we have students that move into our district? Yes. But they pay for their own housing and they pay taxes in that district. We also have all students that live in our district.
I know that there are private school attendees that pay their own tuition, but I also know that there are scholarships that are given to other students. Again, I'm not saying this is a bad thing or wrong, just that the private schools do have an advantage in that situation.