Well nuking the other side eliminates the chance that they'll ever nuke you (or will attack with conventional weapons), so there is arguably a slight positive for nuking first as opposed to keeping the peace.

There were some very serious thinkers arguing for a first strike against the Soviet Union immediately after WW2, including (on some readings) Bertrand Russell, who later became a leader of CND. And a pure CDT (with selfish utility) would have done so. I don't see how Schelling theory could have modified that... just push the other guy over the cliff before the ankle-chains get fastened.

Probably the reason it didn't happen was the rather obvious "we don't want to go down in history as even worse than the Nazis" - also there was complacency about how far behind the Soviets actually were. If it had been known that they would explode an A-bomb as little as 4 years after the war, then the calculation would have been different. (Last ditch talks to ban nuclear weapons completely and verifiably - by thorough spying on each other - or bombs away. More likely bombs away I think.)