In the South, it often did. Private schools sprang up overnight as Lyndon Johnson, working through his bullying satrap Leon Panetta, threatened school officials throughout the South with imprisonment right now if they didn't desegregate their schools right now.
The private schools were legal, often hired teachers who'd left the new Yankee-run school regime (Southern States and counties totally lost control of their school systems during this period, everything was federalized under a cafeteria-style intervention by USG), and were referred to by the sneering liberal mot du jour, "segregation academies".
Nobody has ever tested to see whether those "segregation academies" outperformed their captive public counterparts, which generally declined from the late 60's onward.
Yep and Desoto was started in 1969 I believe.