Does it check out? India and Pakistan both have nukes and keep semi-regularly fighting each other.

MAD "on paper" prevents either side from going far enough to provoke the other into using nukes, but even then it's fundamentally flawed because it works on the assumption that both sides are both rational and believes the other side to be rational, as well as that both sides understands the others red lines well enough.

Already Reagan realised that isn't necessarily true - after Able Archer '83, he realised that the Soviet leadership seemed to genuinely believe that the US might be prepared to carry out a first strike, and that Able Archer got dangerously close to convince them one might be imminent. It's one of the things he noted as a reason to get in the room with them and negotiate.

If you believe the other side is irrational (whether or not that is because you are irrational), and think they're about to strike, MAD turns from a deterrence into a reason to try to preempt to ensure you're the "least destroyed" by hitting harder, sooner.