He wouldn't need to. Taking the Baltics would effectively result in a state of war between NATO and Russia. NATO would be left with no choice but to react.
So if you were the deciding vote in how NATO would respond, and Putin said
"my finger is on the nuclear launch button if you intervene militarily", you're going to roll the dice and call his bluff? You prefer the risk of nuclear war to accepting NATO in its current form is obsolete? That sounds like pride rather than objective thinking.
Why not just re-arm Europe and create a fortified line along the Polish, Hungarian, Slovakian and Romanian borders, accepting NATO over-extended? Yes Lithuanians, Latvians and Estonians would scream betrayal, but they haven't exactly boosted defence spending to 5% of GDP either. How can the lives of hundreds of millions of Europeans, Canadians and Americans be deemed no more important than a few million people living in
"far away countr[ies], .... people of whom we know nothing"?
I can understand the positions of people who think we should not stand up to Russia over Ukraine, but they have no credibility when they say we should draw the line and risk a nuclear war over the Baltics. Ukraine is much bigger it would be possible to bog Russia down around the outer areas away from Kiev. A conflict could also potentially be mostly limited to Ukrainian territory.
Whereas the Baltics are small and would be hard to defend if Russia went all in, as there's only a small land corridor that Russia could easily occupy. It would be very easy for Russia to flood such a small area with men and equipment. Retaking them would almost certainly mean attacking Russia itself, which would mean all-out war across most of mainland Europe, devastating the global economy for decades.
Because the Ukraine is not NATO territory
So? Poland wasn't part of any military alliance with France or the UK in 1939, nor was Kuwait in 1990 with NATO states. History is filled with examples of countries intervening to support another that they have no formal alliance with.
In contrast if the majority of NATO stands by and allows the Baltics to be invaded, there's no legal mechanism to force a military response. The US Supreme Court isn't going to take command of the US military, nor is it going to appoint someone to order them in. The best that could happen would be the President being impeached, but that's not something that would happen in the short window before Russia won.
Slightly different factors/circumstances at play.
It's the same attitude regarding the "reasonable dictator", the idea that someone who is so unreasonable as to brutalise his people and invade a neighbour that was not acting in a threatening way is still reasonable enough not to cross a different red line. There's no reason in principle that Putin could not miscalculate as Hitler did other than we really, really hope he wouldn't because we don't want to die in a nuclear holocaust.
In short I am not saying we should send soldiers and equipment into Ukraine to defend it directly, but I do think there are risks in not doing so if Putin believes it shows we're weak rather than being prudent.