"Civilization would end if MAD was triggered" is a common enough trope but, actually, that mostly means the northern hemisphere, as the nuclear powers. I've not seen anyone mention, even in fiction, what strikes there would be against the regions in the title barring European/US colonies. The trope comes from the Eurocentric/US-centric view of everything, I suppose.
Was there any nuclear power with a sort of "let's drag everyone down with us" kind of MAD mentality? Or would regions be struck to deny it to other powers, such as USSR/USA striking Latin America and Africa to deny it to each other? Especially after the US-led coups in Latin America, I could see the USSR doing something of the sort. And before them, I could see the US doing the same thing as the region did lean towards socialism. Question is, were there plans at all?
If you are asking, was every country targeted with nuclear arms in US and Soviet war plans — no. They predominantly targeted North America, the Soviet Union, Europe, and China (early US war plans did not allow not targeting China). However there were, at various stages of the Cold War, American weapons stationed in Northern Africa (Morocco and Tunisia) and parts of Asia (South Korea, Guam, Okinawa, the Philippines, Taiwan). So those present possibilities for wider targeting.
Would the USSR or US target central/southern Africa, the Middle East, India, or South America? Probably not. More long-range effects of the tens-of-thousands-of-megatons exchange though, such as an impressive increase in global radioactivity through "global fallout" and potential climactic effects (e.g. "nuclear winter") could be indeed global.
If you are asking, is the idea that everyone would die instantly false — yes. That is false even for the Soviet Union and the USA, who even in the most pessimistic estimates would only lose something like half of their total populations immediately in such an attack.
But a world suddenly reduced of huge centers of production and political control would suffer in the years to come as well. Would that lead to a true "collapse of civilization"? I don't think anybody knows that. But the consequences would be unpredictable enough, and large enough, for most people to regard it as "unthinkable."
To your question about whether anyone had a "let's drag everyone down with us" mentality — the only country that has had reported that it had that perspective is Israel, with its so-called "Samson Option," which was a nuclear policy based on the idea of "if you let us get destroyed, we'll destroy a lot more than just us." It has never been clear to me whether this was a true policy or not, or just a form of deterrence signaling/propaganda. I have not heard of any other country with a similar policy. The US and USSR did not have such policies, though at some level any retaliatory nuclear attack starts to have that flavor to it once deterrence has failed.