Racism was literally invented so the working class would fight one another instead of rising up against the elites. Look up Bacon's Rebellion in 1676.
So you're correct, for certain definitions of "working class," but I find attempts to dismiss anti-racism (and increasingly LGBT ) progressive efforts as being merely a distraction from class struggle to be tone-deaf at best, and counterproductive at worst.
Intersectionality is a thing. Race shapes class - especially in the United States where class was literally built around racial caste and identity - and class shapes race. We can have praxis along more than one dimension.