Depends, personally I'd see the number of times the US and USSR nearly went to war (or actually did, but indirectly) despite having nuclear weapons as pretty good indication that lack of any feasible strategy for inflicting a decisive military defeat on the other party was a much more significant factor in the Cold War not turning hot than any nuke-induced desire for peace
Ultimately, one naval officer's minority vote[1] prevented a Soviet sub from launching a tactical nuclear weapon at US shipping. Several thousand miles of ocean prevented the Soviet Union from feasibly achieving decisive military victory against the United States.
[1]https://en.wikipedia.org/wiki/Vasili_Arkhipov
It's true that an invasion of the US mainland was never in the cards, given the situation during the cold war. Pretty much nobody was seriously worried about that. What NATO was very worried about was a Soviet invasion of Western Europe, which is a very reasonable worry considering the number and detail of Soviet plans for doing just that.
For their part, the Soviets were very worried about a combined NATO invasion, which seems quite reasonable considering that Germany alone, while also fighting/occupying most of the rest of Europe, managed to get to the suburbs of Moscow, and kill millions of Russians along the way.
Now if the Soviets did manage to invade and conquer Western Europe and maintain control over it, then they might be in a rather better position to consider direct attacks against the US. Which is a good part of the reason why the US invested so much in keeping them out of Western Europe.
Ultimately, one naval officer's minority vote[1] prevented a Soviet sub from launching a tactical nuclear weapon at US shipping. Several thousand miles of ocean prevented the Soviet Union from feasibly achieving decisive military victory against the United States. [1]https://en.wikipedia.org/wiki/Vasili_Arkhipov