The Americans had already used the bomb, twice, and even before that there wasn't really any doubt they would if they could. As history proved both before and after Nagasaki, the US was happy to cause any amount of death and destruction to fight the (mostly extremely paranoid and overstated) 'threat of communism'. And whatever you think about the idea of nuclear deterrence in the modern era, there's little to no doubt it was the only thing preventing the US using nukes against the USSR.

Presumably by allowing MAD to exist, instead of the ability to destroy the world being solely relegated to the only state who ever exercised said ability in a wartime capacity

The US was feining to drop a nuke in every country they invaded and failed to take over. The nukes in Japan were supposed to be a warning to the Soviets and the rest of the world of who runs the world order now, and the Soviets’ advancement in nuclear technology prevented that from happening, at least for a good while. The overt and covert barbarity shown by the US in subsequent operations showed that it had no qualms about adding more to the list of 200,000.