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.