This is illogical. MAD applies regardless of what Putin thinks or does. I think you meant Putin could presumably go crazy and order a nuke to be used in Ukraine. He could of course, but MAD would still apply regardless. Furthermore, even if Putin were to order the use of nuclear weapons, the probability of the Russian Military actually following through with such an order is extremely small. Though Putin may not care if he lives or dies, that is decidedly not true of everyone else in the Russian chain of command. One can not go through life basing decisions on the highly improbable. As rational beings we are by definition constrained to base our decisions almost always on the probable in preference to the highly improbable. If we make decisions on the latter criterion we are by definition not behaving rationally.If his power is threated MAD no longer applies to him.
There are plenty of irrational people, but they can't, by themselves, obtain and detonate a nuclear weapon, although the media would like us to believe that would be possible. Logic is, of course, not the media's highest priority.