Part of MAD is predicated on concern about the thing being destroyed.
There aren't many countries run by dictators these days, but they exist.
I'd be a lot less comfortable in their making rational decisions not to use The Button. (After having moved their wealth and/or person offshore)
Does someone already running their country into the ground for personal enrichment, and vacationing in places {not their country}, really care about their country?
And well, in that case emissions globally would go down. So climate change would slow.