Much of the problem, of course, was that Southern states didn't just allow segregation, they required it. Plessy was about a Louisiana law that forbade private railroads from establishing integrated cars.
Many of the restaurant owners to whom you refer may have wanted for business or moral reasons to integrate, but Jim Crow prohibited it.
Of course, it didn't really turn out that way, did it? Government got more powerful. Sure, the power was expressed differently, and in a "better" way. But still, government power was (and is) used to mandate social situations in the US. Affirmative Action is just one example where, if the skin-color of the participants is not pleasing to the politicians, then the full power of the federal government can be brought to bear on the transgressors.
That may (perhaps) be better than Jim Crow. But it's not all that different, really. I just think that less government intervention would have allowed citizens and businesses to make their own decisions. Government solutions are rarely good solutions.
The US has by-and-large imposed government solutions to our problems with race relations. How well do you think has turned out?