It is most definitely disputable that it destroy social cohesion. The US was cohesive after WWII, by that point we already had a LOT of non-white people. Also black Americans have been here since the start.
1950 Census
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Race Percentage of Total Population
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White 89.5%
Black 10.0%
Other Races 0.5%
Keep in mind the black population was essentially segregated at this point.
I think it's important to note that our understanding of whiteness has changed over time. And, during that time, there was division amongst even what we consider today as white people. Polish and Italian immigrants often faced discrimination, for example.
If you get rid of race it doesn't fix anything because the underlying cause was never race - it was, and remains, humanities perverse desire for hierarchy and prejudice.
The white people will just then find some other irrelevant bullshit to be mad at each other about and will then segment themselves more. They will literally invent new races and ethnic groups, if they have to, to distinguish themselves from others.
