Of course, none of this means that they weren't inspired by socialism or that they didn't appropriate Marxist ideology for their own aims. They certainly weren't orthodox socialists, but they were absolutely using their playbook. How else would you "negate" leftist ideas? You're also eliding the fact that Mussolini, whose Italian fascism of course inspired German racial fascism, had been a leading socialist intellectual before he was expelled for being too much of a nationalist. Mussolini's fascism was explicitly socialist, in the sense that it was the kind of "state capitalism" that Lenin argued was an advance from liberal capitalism. Whether Hitler was ever an orthodox socialist or not, he was never a laissez-faire liberal capitalist.
The confusion over placing fascism on the American political spectrum is that American conservatism is largely a species of liberalism. Some question whether there even is a conservative tradition, like the one in Europe, in America. Fascism is clearly right-wing in the European context, but it's not clear that it's "on the right" in America, where liberal economics and limiting the state is such a big part of conservative thinking.