I don't think the original post indicated a desire to keep one's children "racially and culturally pure" but rather to avoid sending one's children to schools with large number of non-Whites, for any number of reasons.

It's not a race thing but rather class. People want their children to go to school with отчер kids who are smart / motivated to succeed - hence will focus on school.

Best determinant of that are the parents themselves. So private schools end up being essentially a sieve that checks parents. In the US parents move to the suburbs with high local taxes that fund good schools for the same purpose.