An AIPO poll from September 1950 found that 28% of Americans wanted the use of nuclear weapons in Korea, 60% opposed it, and 12% were unsure. (It was contrasted with the same results from Australians: 18% for, 73% against, 9% unsure.) ^1

A running Gallup poll shows that Americans had a heightened fear of a broader war breaking during the Korean War, with 76% of Americans saying they feared that world war would happen in the next 5 years in March 1951, and the number staying around 50% until after 1954.^2

This poll data (what little we have of it, and what little we know of its methodology — polling was still in its infancy, and even today it frequently gets things "wrong" in terms of being non-replicable and failing to predict certain events) reinforces a generally scholarly view that the Korean War was a general high-point for atomic anxiety in the United States, where lots of people thought that it was highly likely that nuclear weapons would be used and that a broader war (with China or the USSR) would break out.

That does not mean that Americans on the whole were eager for the use of the atomic bomb. Some were, of course; there were those who wondered what the hell these weapons were for if the US could be beat to a standstill in a place like Korea. MacArthur himself was one of these Americans; it was not an uncommon opinion in the military. Truman himself engaged in a little bit of brinksmanship about the possibility, but it is pretty clear that he was not actually interested in using nukes in Korea (or much elsewhere, after Hiroshima/Nagasaki).^3

It would be interesting to see how the polling data changed over time on this question; the situation in 1950 was very different than 1953, for example, and Americans were understandably frustrated back the "back and forwards" of the war which ended up essentially where it began. I suspect you would find an increase of interest by 1953 in the use of the atomic bomb to "break the stalemate," but also perhaps a growing unhappiness with the idea of a prolonged war by itself (the Korean War was never very popular, and the idea of turning it into a large war with China was probably not one anyone but true hawks viewed fondly). But I am not sure we have data about this (I don't have it easily at hand, anyway).

Anyway, my sense is that the song was not necessarily reflecting a general US interest in this idea when it came out. But it was certainly not an unthinkable idea.

[1] Hazel Gaudet Erskine, "The Polls: Atomic Weapons and Nuclear Energy," Public Opinion Quarterly 27, no. 2 (Summer 1963), 155-190, on 181.

[2] Tom W. Smith, "Trends: The Cuban Missile Crisis and U.S. Public Opinion," Public Opinion Quarterly 67, no. 2 (Summer 2003), 265-293, on 277.

[3] See Nina Tannenwald's The Nuclear Taboo: The United States and the Non-use of Nuclear Weapons Since 1945 (Cambridge University Press, 2007), which discusses the Korean War aversions to nuclear use pretty extensively.