I think that proposed equilibrium would have been extremely unlikely under circumstances where the US (a) had abandoned their pre-war isolationist policies and (b) were about to embark on a mission of bending other nations, often through military force, to their will. Nukes had just been used to end a war with Japan. Why wouldn't the US use them to end the Korean war, for example? Or even to pre-empt it? Or to pre-empt any other conflict it had an interest in? The US acted incredibly aggressively when a single misstep could have sent Soviet missiles in their direction. How aggressive might it have been if there was no such danger? I think you underestimate how much of a show stopper nuclear weapons were in the 40s and 50s. There was no international terrorism or domestic activism that could exact punitive measures on those who threatened to use or used nukes.

Even though the cold war is long over, I am still disturbed by how many nuclear weapons there are in the world. Even so, I would much rather live in this climate than one in which only a single nation - a nation with a long history of interfering with other sovereign countries, a nation that is currently engaged in two wars of aggression - was the only nuclear power around.