The Russian leader certainly does care about the risks. However, the nature of the game rewards him for acting that he does not, and the other players for reciprocating. Up to a point, that is. MAD is not a good or stable game.
A close historical precedent for the current situation is the US against Japan in which the nuclear power didn't have a reason to destroy its enemy (indeed, worked to rebuild it immediately after), and the enemy couldn't retaliate on par. So there was no escalation and instead a quick capitulation.
If Russia uses tactical nukes against Ukraine, it matters how it uses them. Presumably it would like to use them as the US did, so it can take Ukraine without destroying it and put a faster end to the war than otherwise possible.
The US can then concede, escalate the proxy war by giving Ukraine tactical nukes, or strike Russia directly. The latter still looks mostly like MAD to me.
A nuclear proxy war limited to tactical nukes is.. unfortunately all too easy to imagine, with some precedent in the Turkish/Cuban missile crisis. That could then escalate until Russia dominates with its strategic nukes, but presumably Ukraine would capitulate before that. The lesson of high cost to Russia would also be there for future reference.
Putin wants everyone to think that, the reality is that this is almost certainly a ruse. I believe it was Nixon that wanted the Soviets to think he was mad and might press the button at the slightest provocation but it was a lie to get them to back off. There’s no evidence that Putin is suicidal. This is a nuclear steering contest which you can only win by not blinking because Putin doesn’t want Russia to be a smoldering ruin with his own bones a melted pile in the middle.
Who’s buying that line after columns of Russian troops tried to blitzkrieg Kyiv earlier in the war? Not to mention that eastern Ukraine is useless to Russia from a strategic perspective. They need to secure the Polish gap, the entirety of the Carpathian Mountains down to the Bessarabian gap, and across to the Baltic coast through the Baltic States. That shrinks the Russian border down to a limited and highly defensible size. Putin has long bemoaned that Russia gave all that up when the Soviet Union split up.
Or maybe he's just better at "madman theory" than Nixon? https://en.wikipedia.org/wiki/Madman_theory
I think you should read up on the Cuban missile crisis.
Until very recently, everyone agreed that nuclear weapons could not be used because the outcome would be that everyone is destroyed.
Putin has ended that era because appears unconcerned by the prospect of MAD.
MAD only works if all sides agree that they do not want to be destroyed. Putin appears to not care.
So, the equation has changed and we all need to rethink the new equations. It may be as simple as - "there will be a nuclear war because a madman has the keys and is willing to press the button".