The experiment was done in the context of Adolf Eichmann arguing at his trial that he was not a Nazi, never was, and was merely following orders. This was of course not even remotely true. If the argument was instead "I only became a Nazi because of their propaganda", that might be plausible but falling for propaganda doesn't make you less culpable for atrocities which you committed willingly.

The version of the Milgram experiment taught to undergrads asks people to believe that you'll follow orders you would ordinarily consider abhorrent simply because you were commanded. But there's basically no evidence for that. People follow orders if those orders are justified in a way that seems persuasive. Nobody ever doubted that Nazis persuaded people to join them. That's not a surprising or even remotely novel finding.