The short of it is that Nietzsche's sister had control of his papers once he went nuts, and his estate after he died. She and her husband were committed anti-Semites, and in her old age, she even was a supporter of the Nazis. Anyways, point is, that she worked to craft his legacy into that of a German nationalist, anti-Semite, which, as you say, is pretty antithetical to much of what he actually expressed in his writings. For longer treatment of this, I addressed Nietzsche's Nazi legacy previously, which you can find here. I'll do my best for any follow ups you might have!

The issue with his sister has already been mentioned, and it certainly plays a big part in the Nietzsche reception. Here, I'd like to focus on some other points.

Nietzsche and his philosophy are (in lack of a better word) complicated. This being a history and not a philosophy subreddit I think in our approach to this question we shouldn't focus on Nietzsche's work itself but rather on the interpretation of it by others. Some people read Nietzsche and recognize an ideology that fits national socialism, others do not. Instead of discussing our personal interpretations here we should in my opion focus much rather on the interpretations of others.

Nietsche interpretation is as complex and complicated as his work itself. People from all kind of ideological back grounds see his work as toxic and unfit, all sorts of other people see Nietzsche's work as fitting to their world view. Just let me give you some facts quickly:

• The German national socialist regime did use Nietzsche for propaganda purposes. His sister very much adored Hitler, but as a matter of fact was not Nazi but deutschnational.
• Some people argue that she was used as scape goat that now takes the blame for everything repulsive in Nietzsche's work; even for those works that were published in his lifetime.
• Alfred Baeumler, next to Ernst Krieck one of the, if not the leading Nazi philosopher saw Nietzsche as somebody perfectly in line with the Nazi world view (Weltanschauung)
• Martin Heidegger praised Baeumler's Nietzsche publishings, yet very much disagreed with the idea that Nietzsche's philosophy reasoned with NS ideology. He very much tried to protect N.'s writings from nazi influence, in the end stepping back from his position as chairman of the Nietzsche Archive in Weimar when he couldn't push through uncensored publishing of Nietzsche's work.
• Ernst Krieck disagreed with Baeumler, too; famously saying

„Alles in allem: Nietzsche war Gegner des Sozialismus, Gegner des Nationalismus und Gegner des Rassegedankens. Wenn man von diesen drei Geistesrichtungen absieht, hätte er vielleicht einen hervorragenden Nazi abgegeben

"All in all: Nietzsche was an opponent to socialism, opponent to nationalism and an opponent of racial beliefs. Disregarding these three inclanations, he possibly could have become an outstanding Nazi."

So even within national socialism, philosophers were divided on whether Nietzsche's work fits the ideology. Outside of it, that also is very much true. There are smart people on both sides of the argument, Georg Lukacs probably being the most famous one on the Nazi-Nietzsche side.

Maybe the German author and journalist Tucholsky put it best when he said:

Einige Analphabeten der Nazis (...) nehmen Nietzsche heute als den ihren in Anspruch. Wer kann ihn nicht in Anspruch nehmen! Sage mir, was du brauchst, und ich will dir dafür ein Nietzsche-Zitat besorgen."

Some nazi illiterats nowadays utilize Nietzsche as one of their own. Who can't! Tell me what you need and I'll give you a Nietzsche quote for use.

TL;DR: Your quoted "massive idealogical divide" of Nietzsche with Nazism is questioned by many. In the end, Nietzsche's work is complex and complicated. You'll find thought that can be interpreted as pro-nazi as well as some being interpreted as anti-nazi.