If you change or move people (e.g., Dr. Martin Luther King), all you do is delay civil rights for a few years. Segregation survived in 1900 because so many people of all races lived in a pre-modern environment; poverty by today's standards. People traveled by railroad and "separate but equal" could be tolerated because people of all races could still get where they needed to go.
To retain segregation today, you would have to butterfly away most automobiles, cinema and television. In other words, no twentieth century progress. When did South Africa finally get television? I think it was after 1970. Apartheid simply does not survive in a modern, quasi-free society.