How did American “wokeness” jump from elite schools to everyday life?

This. It's amazing what you find in the shadows, in the periphery of our culture, when you simply look. For a long time, we just refused to see what was right in front of us. There's a long tradition of transgenderism, and anthropologists have been writing about it for a while.

The NSDAP was about as left-wing as Tucker Carlson.

There seems to be an ahistorical reading popular with especially Americans today that because the NSDAP called itself a "worker party" and had some left-sounding campaign promises (though mostly in the 1920s) it was a "left-wing party".

Germany had been a monarchy until the end of the first world war. "Right wing" in Weimar Germany meant "monarchist", not "free market". The monarchy was overthrown by a left-wing coalition that rapidly split into the more centrist SPD (anti-monarchists who just wanted a free republic but not challenge the social hierarchy itself) and the various socialist tendencies (anarchists, Trotskyists, Marxists, Bolsheviks, etc). The SPD also famously allowed the (monarchist and right-wing) Freikorps to kill one of the most influential socialist groups. That's all before the NSDAP was even a thing.

The NSDAP started out as a fairly uninteresting nationalist workers' party. It went through several iterations and multiple names that increasingly brought a focus on nationalism and a hyperfixation on an imagined betrayal by the leftists causing the defeat in the Great War that ultimately became part of the Jewish conspiracy theory and the idea of Cultural Bolshevism (which you may know as "Cultural Marxism"). To the NSDAP the clear enemy were the communists, the Bolshevists, but in general "the left" including the SPD. They continued using leftist rhetoric in some of their material for a while because it worked but much like Tucker Carlson ranting about coastal elites and big corporations their answer wasn't to dismantle capitalism or tax the rich.

In the end, the NSDAP heavily vilified the communist DKP and the left-wing SPD as well as the "international Jewish bankers" (who in their mind unlike the "good, German bankers" were a nomadic people in a profession they turned parasitic by extracting the wealth from the German people and bringing it outside the nation - so to them it wasn't capitalism itself that was the problem or even banking). The right-wing conservatives liked this and even pardoned Hitler after his failed coup because they thought they could use the NSDAP to crack down on the still growing socialist movements and prevent a second communist revolution like in Russia. The Enabling Act was signed by the Christian conservative "center party". The plan worked, though not quite as intended. They did however crack down on unions and many of the first people sent to the camps were socialists and queers. They also nearly invented the idea of "public-private partnerships" although they wouldn't have called them that.

Party politics change over time and thinking that "the left" must always have been what the Democrats do and "the right" what the GOP does is not only extremely centric to American history but also ignores that even today the US political spectrum has more than two parties (not to mention the vast differences between local chapters) and none of the parties can or could ever be neatly summed up as "left" or "right".

EDIT: Before someone feels like they have to point this out: yes, the SA in particular (the security volunteers of the Nazis who went on to become their paramilitary street gang) had some people with actually somewhat leftist ideas in them and some members of the NSDAP were openly gay. Upon rising to power however the NSDAP purged (i.e. murdered) most of these people. But even the "leftist" ideas tended to be hypernationalist and be tainted by their social views (e.g. eugenics, anti-Semitism, Aryanism and homophobia). This is in part why most leftists today reject any "red-brown alliance" out of principle.

EDIT2: Since some people are prone to misunderstand points made in lengthy replies: I'm not saying Tucker Carlson or the GOP would have agreed with the NSDAP. I'm saying neither Tucker Carlson, the GOP nor the NSDAP are or were left-wing and that leftist populist rhetoric can be found in right-wing politics to this day so judging the NSDAP by a small subset of its rhetoric is historically illiterate.

What's your actual goal with this comment?

I'm really suprised, it has spread around the whole western world. There is not enough pushback on the foundational philosophical flaws of this phenomena. The biggest flaw of "wokeness" it doesn't try to combat the human proclivity to treat other groups of people as sub-human if they aren't members of their own the "correct" tribe. A couple of days ago I just took a break of the internet for the simple reason that people are celebrating the death of anti-vaxxers. Poking fun of people that are so desperate and scared when they get sick so they use a drug intended for horses. Very few people say anything against this and those who do are dismissed as "anti-vaxxers". Wokeness doesn't try to combat this phenomena, it tries to stoke it by forming their own tribe and treat everyone outside tribe as idiots and uneducated buffoons. They don't see that they are feeding a psychoological mechanism that partly the basis of the social ills that arise when we treat other people as less than themselves. "All animals are equal but some are more equal than others" - George Orwell.

But that's not the whole story. Kids are also very malleable, open to peer pressure and easily assume identities that earn them acceptance by their peers.