It is the nonpublic schools that accept most applicants, without regard to where they live. It is the public schools that accept only those they must. There is a great deal, as has been cited here before, as well as generations of experience, but opponents of school choice ignore or deny it. For example, lecturing is the most common teaching method. No research defends that.
Many practices directly contradict research.
A prime example is the building of large schools, often with thousands of students. As long ago as 1964, a review of the research found hundreds of studies had concluded that schools of a few hundred are better - in the range of 200 or so at the elementary level and 600 or less at the secondary. This certainly does occur. It isn't private schools that have boundary or residency lines within which one must live to be eligible for attendance. As was pointed out above, those left behind are the minority, the poor, the otherwise disadvantaged, who are not only segregated in their own school districts but often segregated within urban districts because district size and residency patterns prevent busing students all over the city.