Plus, actually getting rid of all the nukes would make warfare more attractive, which makes acquiring nukes desirable again. And no nation under threat of total war would rather lose everything than use nuclear weapons. So I don't think the equilibrium of nobody having nukes is at all stable: eventually one country begins to think about warfare, decides they need nukes, creating a power imbalance, causing others to get nukes too.
Though of course the flaw of MAD, or even game theory in general, is that it assumes everybody will be a rational actor. People are people, and if someone stupidly violated MAD against their own self-interest or reasoning on some bizarre hundreds/thousands of years timescale, we could all perish. There is a small but nonzero probability on every single day of our lives that we and everything we know will be destroyed by nuclear weapons. So far that day hasn't come yet, and there's no way of knowing how likely it is. But given that banning nukes on an international scale seems unstable anyway, I guess this is the best we can do.