Racism was certainly one of the things that precipitated the shift in the equilibrium. Remember, at the time when people still lived in the cities, the suburbs weren't like they are today. There were good schools and decent housing in the cities, because the middle class white people hadn't moved away yet. As someone pointed out elsewhere, the availability of cars to make the suburban exodus happened long before the suburban exodus itself happened. And let's face it, race place a huge role in living choices today. Check out:
http://www.city-data.com/forum/chicago. People are perfectly happy to be egalitarian right up to the point where it involves sending their kids to schools with heavy black populations. "Gentrification" is almost synonymous with moving blacks out of neighborhoods, and cities like to tout gentrification statistics, with their underlying tone of racism, without any of the bashfulness that is usually associated with even the hint of racism. People in Chicago considered it a great thing that the South Loop had gone from 35% white to 75% white in just one decade.
Look at it from a different perspective. Why is it that in Europe more of the population of the various metro areas lives in the city than in the surrounding suburbs, as compared to the U.S.? Do Americans just inherently like big houses and large lawns more than Europeans? And if they do, why didn't they move out of the city long before 1960, when it was eminently practical to do so?
That of course doesn't mean that suburbanites are racist today. After decades of suburban exodus the core cities are shells of their former selves, urban public school districts are almost exclusively for the poor, etc. There's lots of non-racist reasons, today, to not want to live in the city besides the inherent availability of lawns, etc. But given how racist society as a whole was in 1960, I think it's ridiculous to claim that the racial tension precipitated by the civil rights movement and desegregation didn't play a large role in tipping the balance between the cities and suburbs.