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The KKK aren’t anti-capitalist. The Nazis aren’t anti-capitalist. The Confederates weren’t anti-capitalist.

Except all of these groups were, although the Nazis were much more explicit about it- that is indisputable so I'll focus on the others. The Confederacy was simply a plantocracy near oligarchy, not to mention slavery which is incompatible with capitalism(already explained in another comment, but a brief breeze over it- slave masters act as entities of the state by the very nature of them having a monopoly on regulation of other humans). As for the KKK, much of what it and other racist organizations of its era did was try to "protect white jobs", and lobbied heavily for state intervention to that effect- such as targeting immigrant and black "scabs" and pushing for minimum wages aimed at driving them out- though the KKK also explicitly opposed a lot of unions and other organized labor activities(often because they weren't white enough or included Catholics). Fundamentally though, the KKK viewed their goals of white-protestant supremacy as greater than an economic system, and were more than happy to destroy private property and private individuals- or use private property when it benefited them. Similar to what the Nazi's believed- its private profits are okay as long as they are working in the interest of the greater goal, but the second its not they're more than happy to steal it and kill you. Capitalism doesn't require private profit from the means of production, it requires private control of it- and if it can be seized if not following exactly what the state wants, that's not private control.

it is a coherent idea to create collectives within a system that is philosophically oriented around the individual power of property-owners, that’s what labor unions are.

That's what unions are to some people. To other people unions are a convenient organization of people with similar and/or parallel goals on a specific matter(and not necessarily others) so that by collaborating they can achieve their individual goals.

The unavoidable fact of capitalism is that it relies on pushing most of society into the same general social class (workers, as contrasted with owners; employees vs employers)

No, that's not true. Capitalism doesn't ascribe the distribution or organization of labor, just that it is privately controlled. A society of independent agrarian farmers could still be capitalist, or a commune of people who voluntarily donate their labor to each other.

but it’s no less coherent to draw lines of common interest between them, most often something like race or religion that is convenient to capitalists, because the capitalists can say “look, I’m white (or whatever) too, I’m on your side!” even if they truly aren’t because they are only seeking their own profits.

Yeah no doubt, though I think it is a little perverse to use "capitalist" to refer to owners/employers when they themselves are often not ideological capitalists, although it is still a correct use of the word I think it leads to intentional confusion(though not by you, just in general).

The reason, the real motivation, for the racial terrorism I described above despite merely deflecting worker ire, is that by “clear cutting” more space open for your market share by slaughtering competitors, people sitting on land you want, etc., you can gain more room to grow, although this too only lets you grow temporarily until you bump into your new limits, so you need to keep killing inconvenient people to keep growing, and that’s more or less how the Nazis worked, both internally by picking out minority after minority, and externally with their continuous invasions and “lebensraum” and so on.

I don't think business owners are that generally competent to have orchestrated the total destruction of black and jewish owned businesses, I think the Nazis and KKK were both more than motivated enough to do that themselves, but I agree there definitely were some to supported it when they saw it happening and benefited from it.