That wasn't the thrust of Lindsey's argument. She argued that the way Mel Brooks criticized the Nazis was in a manner that made them seem foolish or stupid. Nobody is going to watch The Producers and be able to walk away with any sort of positive impression of Nazis. Whereas in a movie like American History X, even though it's got an anti-Nazi message, the audience can still away thinking that Derek Vinyard was a badass. Another example she used was a Halloween episode of South Park where Cartman's ghost costume resembled a KKK outfit and when Chef saw it he beast a hasty retreat. The problem was that the Klan wasn't the butt of the joke there. Brooks' jokes about the Nazis worked, according to Ellis, because they made the Nazis look stupid, weak, and not at all badass or powerful.
In the video I watched, at the end of it she said exactly what I said - that Brooks could tell Nazi jokes, but other people should not. I remember it clearly as I'd agreed with most of the video - which included stuff like what you just said - and this final bit was shocking (to me) and something I strongly disagreed with.