NATO doesn’t respond, proves itself useless, and dissolves. Putin divides and conquers Europe, marching his army all the way to Portugal. Putin, emboldened, launches an attack on the US. Nukes fly. Game over.

That's assuming the EU won't respond, or for that matter Poland being incapable of pushing back Russia all by itself. There's about exactly one single reason why the Poles aren't parading on the Red Square right now: Because they're in NATO, which acts as a leash. Baltics pretty much have the same attitude but are smaller so they'd simply follow Poland. Finland would get pulled into it because of their own attitude and Estonia, and with them, without fail, Sweden. At which point Germany would have a hard time holding back and then it's guaranteed that the French will be in the fray, and that's presuming they wouldn't have been as soon as Poland lets loose because principle.

Now the US in its usual exceptionalism might be blissfully unaware of those dynamics, and the Kremlin because the FSB reports what the Kremlin wants to hear, but it's true nontheless. But in the end once the EU is involved the US will be, too, because the US can't countenance Europe doing something militarily without joining in. Reluctantly and in a limited fashion, probably, just as they're reluctant now. Germany has pretty much stopped trying to bully the US into providing more things because we've reached the limits of what the US will do (that is, Germany could pressure the US to deliver Abrams by tying Leos to the US also delivering tanks, but providing Taurus cruise missiles won't be tied to ATACMS because apparently that's a US red line).

We’re not sending anyone to their death. Ukrainian soldiers are doing the fighting, not American ones.

Mask off moment. Ukrainians aren't even people to you. Most of them are conscripts forced to be on the front line.