Schools are segregated due to quality of life issues. Blacks and Hispanics are associated with crime and school violence. Ask Asian immigrants who come to the US. The ones that go to majority black and hispanic schools receive daily beatings, lunch money robbery and tauntings from mainly black and hispanic students. Most Asians find the blacks are the worst while the hispanics are the least. Contrast the experience of Asian immigrants who go to majority white schools. Attacks and harassments drop drastically. So based on that experience, guess where most first generation American Asians are going to live and send their kids to school?
Schools are segregated due to quality of life issues.
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Private schools are far less segregated than government schools.
And...Private schools are less segregated in subtle ways as well. For instance, when lunchroom seating is studied, children in private schools are far more likely to sit at tables with mixed races of black children and white children. In government schools children in the lunchroom segregate themselves into white tables and black tables.