There's no WWIII if we appease Russia. There's a lot of other bad things though, including allowing Russia to get another win and to put up with their influence in our politics and society and watching as we slide further into Fascism. But that isn't a nuclear exchange.
We are risking WWIII by confronting Russia because of all those other things.
The situation is precarious enough that Biden has threatened retaliatory decapitation strikes to kill Putin in the event of the use of Russian tactical nuclear missiles. And that sounds like a rational and proportionate response that might avoid nuclear war (but opens a massive can of worms since retaliatory assassinations become likely).
You can't "flip the script" and argue that its the other side of the debate that leads to nuclear annihilation. You don't get off that easy, you have to deal with the fact that its worth it to risk it.
(Although with the way our politics is going it may be ultimately immaterial anyway in the long run).
Ah, no actually? None of that. Obviously he won't launch nukes after he gets what he wants. He probably won't even launch them if he doesn't, though that counts on him keeping a shred of rationality.
My only point here (distinct from those of others on the chain, please note) is that preserving liberty globally depends on not being cowed by someone threatening to use nuclear weapons. Even if they never actually launch, it would be a catastrophe if aggressors could get away with anything just by threatening to use nukes. Without exaggeration, you're potentially talking about the death of liberty worldwide. The only reason this is unlikely is that free nations tend to wake up before they get completely boiled frog-style; this is, in fact, my read on why everyone is suddenly supporting Ukraine and cutting off Russia after doing jack shit in 2014, they've lost all deniability on the threat Russia poses.