I agree with your post. The Nazis certainly were left-wing socialists. As were Mussolini’s Fascists. Your reasoning is solid.
Some people classify the Nazis as right-wing simply because conservatives support a strong military, and so did the Nazis. That argument is both superficial and wrong.
But I will add one thing. There can be right-wing fascists (small f there). Such people believe in property rights, individual freedom, etc. in theory. But they will not tolerate any disagreement or dissent. That’s fascist behavior.
What part of "National Socialist" do they not understand?