That might have been the case. And what did white's do? They obviously went to the suburbs. They left the inner cities.
I remember reading something Sherman once said, basically 'he' had to kill two hundred thousand or so rebs, for their own good, to save them.
Things come in cycles, at the turn of the 20th century, America was being divided with immigrant crimes. Unfortunately, the Gov't has divided so many people that things will fall apart when the goal is to bring everyone down.
Well, that conflates white response with another social trend that began before WW II, suburbanization, but your point is taken, and I noticed when I was reading The Making of the President 1964 that Theodore White had to devote quite a lot of space to the phenomenon of "blockbusting" and black crime, and also rioting. That was the year that there were, for the first time, some serious disturbances that became the 60's race riots.
I think a case could be made that crime was only a part of white flight, and that black political control (and the white surmise that nothing would henceforth be done about either "black crime" or encroachment against white living spaces, with black politicians protecting the offenders) was the key, as in Detroit, to white abandonment of some cities. This didn't happen in Chicago or even New Orleans, n/w/s that both cities had and have corrupt political regimes. The New Orleans political regime has been exclusively black and overtly racist since the mid-70's; no white can get elected to citywide office, and the last white councilmember to question the corrupt appropriations of Marc Morial had ghetto blacks from the project bussed into her district to vote her corruptly out of office.