I've thought about this question a lot since last night, and I'm not so sure anymore that nukes would guarantee peace. If a country (or government) was facing an existential crisis (as Ukraine's currently seems to be), then, yes, perhaps. That country would be more willing to deploy nukes, and nukes would scare away that particular invader.
But what happens in the case of a limited military action that triggered NATO's Article 5 — if, say, Russia tried liberating Estonia's Russian population? Would we really be going to DEFCON 1, or would the Internet be debating the worth of destroying civilization for the sake of 300,000 people (which, incidentally, would also destroy Estonia)? Even if Russia straight-up conquered Estonia (with a small population of 1.3 million), my hunch is that we'd still be debating the merits of launching nuclear weapons.
So far, the West has been supplying Ukraine with weapons in open defiance of Russia. No nuclear war has resulted.
It's not clear that conventional warfare is going away because of the threat of nuclear war, which means that Putin in fact has a case, and that NATO's geographical encroachment really could pose a credible threat to Russia's defense imperatives (although, after this week's events, I can see why people might want to ignore or straight-up violate them, or to regard them as being offensive rather than defensive in scope).
There is even the possibility that we'll develop the means to neutralize nuclear weapons someday. Point defense systems against missiles are still ineffective. Israel has Iron Dome, but there are reasons to be skeptical about that system —
An MIT professor evaluated Iron Dome and believes that it is over-hyped (with an effectiveness of 5% or below against Hamas rockets in 2014). But that could all change one day, which is something that Russian strategists do have to take into account.