In regards to LGBTQ+ people, Nazis had it much easier and had no need to evolve. The "morality laws" they enforced were already 60 years old when they took power, and had at various times before been applied very strictly. Things didn't change all that much for LGBTQ+ people when Nazis took power - they were shitty before and stayed shitty (unless they were blue-eyed and blond-haired, then conditions sometimes improved a little) - and in fact some queer people locked up by Nazis were required to finish serving out their sentences after Nazis crashed and burned. It wasn't until the late 1960s that Germany (and most of the world) looked back at Nazi treatment of queer people as a problem, and Germany didn't fully overturn the 1870s "morality laws" that Nazis enforced until 1996.
Nazis didn't really go through an evolution on LGBTQ+ people, Germany was locking them up and disappearing them before Nazis, so they were more of a maintenance of the status quo. Today's reactionary Republicans are VERY different, in that even when the USA didn't have the modern protections for queer people it's long been understood that they're not criminal and we don't disappear them. Unlike Nazis, today's Republicans are actually going through a regressive evolution.