Its soldiers saw wide ranging combat experience over the 3 years in
conflict against veteran soldiers of Germany and Japan.
I believe you have answered your own question: The USA was war weary and looked forward with optimism to a time of peace. "The Red Scare" was simply that - a scare - nothing more. The country was trying put itself back together: There was very little interest or motivation on the part of those combat scarred veterans or their families to shove off to the distant shores of China and Korea and fight a nebulous red chimera in another brutal war. In short "motivation" was the indeed the determining factor.
It was not a question of not being able to win, but of not wanting to devote the requisite blood and treasure victory would have required. Unlike some other nations, ruled by despots and military leaders, in the USA the domestic political climate directly impacts foreign policy particularly with respect to wars: The President and Congress are in charge of waging war, and they are civilians, who owe their power directly to the civilian electorate. If the electorate objects to a war, the elected officials will very soon take notice. This pattern has been repeated many times in the course of American History, back to the USA's earliest days.
Initially, the USA was very wary of becoming engaged in a ground conflict in Korea: At the same time, the Administration was worried that a war in Korea could quickly widen into another world war should the Chinese or Soviets decide to get involved as well... The Administration still refrained from committing on the ground because some advisors believed the North Koreans could be stopped by air and naval power alone.
But the Korean conflict was becoming increasingly protracted, complex and deadly - the first of the modern American "quagmires": There were serious setbacks in the fighting around the 38th parallel in early 1951, and the USSR had started getting involved in the Spring of 1951. At that point, Truman himself, who had gone into Korea (under the cover of a UN 'police action') hoping the the US could limit its involvement principally to air strikes and some naval action, became pessimistic about the situation in Korea, as the prospect of a large scale ground war loomed imminent - something he never wanted, and knew would be be politically unsustainable: MacArthur threatened to destroy China unless it surrendered. While MacArthur felt total victory was the only honorable outcome, Truman was more pessimistic about his chances once involved in a land war in Asia, and felt a truce and orderly withdrawal from Korea could be a valid solution. Certainly, the USA could have used nuclear weapons (as McArthur suggested) to settle things, but in Truman's mind, that was the last, worst option, both internationally and domestically.
In fact, the failure to quickly resolve the Korean Conflict was a factor in Truman's decision not to run for President in 1952 (the 22nd Amendment did not apply to Truman: "...But this article shall not apply to any person holding the office of President when this article was proposed by the Congress..." the 22nd was proposed in 1947, ratified in 1951 - all while Truman was President) - his popularity had sagged in part due to the situation in Korea, not unlike the situation Lyndon Johnson faced in 1968, when the conflict in Vietnam was so problematic for him, due to its domestic unpopularity, and was a factor in his decision not to run for re-election at that time.
It would have been impossible for Truman to muster the political support necessary for an extended full blown war against China and possibly the USSR in such a domestic political climate, simply because of a "scare" on the other side of the world, and Truman knew it.
In the presidential campaign of 1952, Dwight D. Eisenhower, who subsequently emerged victorious, included a promise to end the war in Korea: Many of his radio and television commercials discussed topics such as... ending the war in Korea. i.e : It was an unpopular war.
UN intervention and Armistice, engineered by Eisenhower's administration, was really the only viable option.