MAD is a specific sub-doctrine of the broader concept of deterrence. It doesn't meant the same thing; it means a policy that relies on mutual vulnerability. It was only specifically claimed to be adopted by the United States in the early 1960s, briefly. Before MAD there were other doctrines in place, like Massive Retaliation (early Eisenhower), New Look (later Eisenhower), Flexible Response (later Kennedy and Nixon), etc. These are not the same thing as MAD; e.g., Massive Retaliation means that the US was prepared to use massive, disproportionate amounts of retaliation even for conventional excursions that it deemed to be crossing a line; Flexible Response meant that the US was willing to engage in limited nuclear war. A true MAD approach would also avoid things like missile and bomber defense, because you are trying to emphasize the aforementioned mutual vulnerability.

I point this out just because for most people, they confuse MAD with deterrence broadly, and it is a different thing. All of the above are flavors of deterrence theory, which is based on the idea that one state can deter another from doing something it doesn't want them to do (say, attack them, or invade West Germany, or whatever) through some kind of threat to something they value. But there are many ways to threaten without it being mutual.

Anyway, if we re-word your question: "is it fair to say that nuclear deterrence, etc. etc." — the answer is: it is something that is frequently said, but it is actually much harder to prove than most people realize. The existence of nuclear weapons certainly changed the calculus of international leaders in the Cold War regarding conflict and war. (But so did many other things, like the United Nations, the shared experience of World War II, increased economic cooperation between nations, etc. — nuclear weapons were not the only big change that happened after WWII, is my point.) Was the fear of retaliation the major reason they didn't engage in certain types of aggressive activities, and engaged in others instead? It is not a ridiculous inference, but to prove this you'd need to be able to get inside their heads and planning decisions in a way that is very difficult to do, and you'd need to show — somehow — that if nuclear weapons didn't exist, they'd have done something else. Which is such a large change to the overall context that it becomes a very speculative exercise.

Similarly, one can wave one's hands and attribute proxy wars to the existence of this dynamic, but it's a very high-level counterfactual to say, "if nuclear weapons didn't exist, then the Soviets would never have invaded Afghanistan." It is much easier to point to the specific causes of the Vietnam, Korean, Afghan, etc. proxy wars and talk about the more proximate conditions did lead to them, rather than the broader framework.

Again, this isn't an answer of "no," but it is an answer of, "this is a more complicated historical claim than most people realize." It is also important to note that the major powers came very close to open conflict several times, and that nuclear weapons did not avert direct conflict between, say, the Soviet Union and the People's Republic of China, but it may have muted it. And there are other complications — like, why nuclear weapons weren't used by states that had them even in situations where nuclear retaliation wasn't a threat.

Frequently when these kinds of claims are deployed, they are done in the service of some sort of bigger political point — e.g., nuclear weapons were a net good, because they prevented open conflict and relegated it only to "smaller" conflicts (which nonetheless claimed millions of lives, I think it is important to add). In such cases, it is not really meant as a robust historical argument. Suffice to say, as both a historical argument and a strategic argument, it's a pretty complicated issue if you take it seriously, as opposed to just using it as sort of a slogan.

Anyway, I think it is fair to say that lots of people have claimed this over time. I think it is a lot harder to make the point in a rigorous way.