In the late 1950s and early 1960s, despite intentional or unintentional information to the contrary, we know that the United States had a great superiority and reliability in ICBMs and other delivery methods compared to the USSR. Obviously it would be assumed that some damage in the US would occur in a nuclear war at the time, from a couple of destroyed cities to dozens of cities destroyed and tens of millions dead. However, ultimately this would not have been mutually assured destruction, since the US could eliminate virtually 100% of the USSR with high confidence. At what point (month and year or year) would it be considered that America could simply not survive a Soviet attack, and thus the two sides would virtually become equals in this scenario? Furthermore, forgetting the obvious moral reasons, and thinking about this in a strictly rational way, would it not have made the sense for the US, if they knew the stockpiles would grow to endanger the entire human race, potentially forever, to strike the USSR as soon as possible in that early-mid cold war time frame? You would destroy a country, but the threat of doomsday for the entire globe would almost certainly vanish. Again, from a purely strategic point of view, as if robots made the decisions, I'm not suggesting it would have been OK to do that.
You could set up a shifting timeline with respect to the Soviet Union's capabilities with regards to the United States:
1945-1949: US nuclear monopoly; Soviet Union has zero capability for using nuclear weapons in retaliation (though it has many conventional forces which could cause pain in Europe and Asia if deployed).
1950-1959: Beginnings of the Soviet capability to threaten Europe. By end of the 1950s the Soviet Union can totally destroy Europe with short- and medium-range nuclear weapons. But they had very limited capabilities (if any) to threaten the mainland USA, and the USA had an overwhelming advantage (tens of thousands of nuclear weapons by the end of the decade).
1960s: The Soviet Union starts the decade by being able to threaten a few American cities credibly with ICBMs. By the end of the decade they can threaten thousands of cities. This is the period in which "MAD" was coined (by McNamara) and in which the US "advantage" gradually decreased. But even during the Cuban Missile Crisis, while the USSR did have the capability of threatening many US cities, the US had a 17-to-1 advantage in capabilities. Does that mean much? In practice, no; the US politicians felt that the Soviet capability to destroy the US and its allies was sufficiently good enough to warrant a credible "mutual destruction," even if many Americans would have survived (the government probably would not).
1970s onward: By the 1970s the USSR has essentially equal capabilities to threaten the USA; true parity has been achieved in both numbers and capabilities. The destruction would be truly mutual (in the sense of equal), and more or less total.
For your followup question, the question of why the US did not use nuclear weapons is a complicated one. Part of the answer is that the costs would have been extremely high even before they became totally suicidal. The USSR had a credible ability to threaten America's European allies, even if before they had the ability to credibly threaten the USA itself. Does that necessarily have to matter? No. But it did, at least to the politicians in charge.
There are other dimensions as well — e.g., the political will to carry out a first strike attack would have had to be immense, and as it was, it didn't really exist — but it is just important to emphasize that deterrence is actually much more than "total destruction," it is really just "unacceptable costs," which is a lower bar than that.