Proliferation and stability are linked, but not a simple causal relationship. For example, look at the Cuban Missile Crisis. It wasn't even 'proliferation' just the addition of new attack vectors (namely, the much increased potential of a decapitation first strike against the USG) that nearly sent everything ablaze.

Basically, MAD only works when both sides are relatively symmetric. When you have weird asymmetric situations, and inability to actual cause true mutual destruction, especially when mixed in with certain ideological elements, nuclear weapons can be anything but stabilizing. For example, an enemy that is already willing to use suicide bombers, or trade civilian/innocent lives to counteract the opponent's advantages may not be fully deterred by nuclear weapons. Especially an enemy that happens to be a non-state actor.

You are of course correct - we do need the long view to actually tell. But I strongly believe that simply proliferation of nuclear weapons, especially in situations without guaranteed mutual destruction is likely not stabilizing.

"It seems like the proliferation of nukes has resulted in a more stable worldwide society."

I think that holds only if all parties that have nukes believe they have more to lose if nukes are used than if not.

Ideology can change this decision. That's why nuclear proliferation remains a dangerous thing.

There have been many significant wars in the last few decades, including many proxy battles between nuclear powers. So I'd say your thesis is disproven right now without waiting.