The US spent nearly the entirety of the post-1945 period contemplating what global conflict would look like, taking it seriously, planning to wage it. There were exercises to simulate it, there were constant developments of weaponry to get an edge in it, there were vast systems developed so that said conflict could be started on the order of minutes. You have to remember that the US official mindset, rightly or wrongly, believed that the Soviet Union was bent on world domination and expansion, and they saw evidence to reinforce this wherever they looked (Eastern Europe, Latin America, Southeast Asia, Africa, etc.) over the course of the Cold War. They believed that the Soviets were willing to risk it all, did not care much about their own civilian casualties, and were just waiting for the appearance of vulnerability to start a massive attack. In some circles in the US it was believed that said global conflict was not only possible but inevitable, and some of the people who believed that even contemplated that it would make more sense to preemptively start said war, while the US had advantages, to get it over with.

Fortunately none of this came to pass, and we now know that the Soviet mindset was more nuanced than how it was imagined at the time. But one should not take the fact that it did not happen, and that the consequences would be so insanely high, as evidence that it was not taken seriously. Only in the post-Cold War did the US really start to believe that said global conflict was unlikely to occur, and even then, new powers (e.g. China, a revanchist Russian Federation) have replaced the USSR as the locus of possible conflict.

If you haven't read Schlosser's Command and Control, I highly recommend it. There are many people today who find it inconceivable to imagine that all of the weaponry and gruff talk in the Cold War was deadly serious, and that these countries were ready to kill hundreds of millions (at a minimum) of civilians in order to wage this ideological/economic conflict, but they were. Schlosser does a good job of painting that picture.