If Putin is that insane (i.e. doesn’t care about MAD) then a nuclear war is pretty much guaranteed regardless of what NATO does.
Again, that's incorrect, see below.
The NATO treaty prohibits acts that violates the UN charter. Article 51 of the UN charter states that UN members have the right to individual and collective self defense in the case of an armed attack. They have this right until the Security Council has taken action to maintain peace and security. In the case of Russia's invasion of Ukraine, the Security Council has so far been unable to act due to Russia's veto, which means that NATO is free to act in the aid of Ukraine's right to self defense.
There's nothing else in the NATO treaty that would prohibit NATO from aiding Ukraine. The reason why NATO hasn't intervened (yet) is because they don't want to escalate the conflict - not because they are prohibited to act.