I’m a sucker for this kind of speculation. Plessy was about the state requiring a private company to segregate. But, the reasoning for and against applies to government facilities as well. In particular, it would have led to outlawing segregated schools. The popular push-back would have been huge.
I want to believe that Americans would have just said “Yeah, now that I think about it, segregated schools are wrong too.” But, I don’t think that would have happened.