Actually, Fraktur was banned in 1941 and the regime transitioned to Latin writing (Futura was used quite extensively).
But it wasn't all government; newspapers (most notoriously Der Stürmer, but many others), street signs, shop signs driving licenses (still valid today) and such were increasingly written in Fraktur.
Then suddenly one day in 1941 it was suddenly declared „Judenschrift‟ -- "jewish writing", and this time a writing was declared legally forbidden. And as far as I can tell, nobody knows why!
Despite all that I have found in my mother in law's house books in Fraktur from the 30s, 40s, and even a couple from the 50s. I have one from the 50s on my desk right now! And you do see it on various signs in shops, some street name signs, and other random other places, especially when you want an old-timey feel (traditional restaurant, perhaps Glühwein Bude in the woods, and such). If people had a strong nazi identification I don't think you'd see it. Maybe that's just a USA thing. Fraktur was used extensively in other parts east of Germany (e.g. Czech) well into the 20th century as well, but I haven't seen it so much there -- I wouldn't at all be surprised if that was due to the Nazi past.
Fraktur has some nice ligatures, including ch, ck, ss, sz, ch, sch, tz, etc, of which only ß (and ä, ö, ü of course) survived in roman. Weirdly, ß is pronounced "s" "z", written as a ligature of s & z, but when unavailable, is replaced in Latin text with "ss".
As for ʒ, it is literally a tailed z, not s, as even your link says.
Not quite: ß is nearly always a lowercase-only letter, and it is capitalized to SS even in German text, where they have all the German letters. A capital ẞ exists but is vanishingly rare:
https://www.thelocal.de/20170711/after-a-century-of-dispute-...
https://en.wikipedia.org/wiki/%C3%9F
Example of capital ẞ from 1957 in German text:
https://en.wikipedia.org/wiki/File:Eszett_Leipziger_Duden_19...
Generally, regarding the aesthetics of the regime, the party line favored the plain, straight, ortholinear as opposed to skewed aspects or the overly ornamental. However, this was subject to favorism and various factions in the regime. (E.g., while the (in)famous shots of Leni Riefenstahl's films would have been in violation of these aesthetic principles and thus should have been deemed "undeutsch", these became pretty much the show case of the regime.) Again, I'm not sure, if there's a connection, but it may provide a hint for why Fraktur fell out of fashion with the regime.
Regarding the "ß" ligature, this predates Fraktur and wasn't specific to German writing. (E.g., there are 15th century Italian examples.)