The American public wanted unconditional surrender by the Axis and achieved it. Meanwhile, Korea was easily defeated and occupied by Japan for decades.
Why did the US public accept defeat by such a weak power when they strongly believed we were the most powerful military in the world? Prior to the Korean War, the US had never lost a war.
(Yes, China assisted North Korea later in the war, but China was also easily defeated by Japan and very weak.)
The United States war aim was to prevent the communist conquest of South Korea (see related UNSC resolutions) without escalating to the point that war with the Soviets, war with the Chinese outside the peninsula, or any form of nuclear exchange occurred. That war aim was achieved. I don't understand your claim that the United States lost the war.
The UN lost Korea? Given that the government that we supported still exists, the borders still exist, and the war was fought to preserve South Korean sovereignty, saying the UN forces lost Korea is just really, really weird.
Well first of all it's sorta hard to say anyone has "lost" the Korean War, due to it still technically going on. Second, the only aim of armed response to Korea was the retaking and safeguarding of the Southern government (ignoring of course MacArthur and a few others who totally wanted to push to china), and last i checked Seoul is not part of North Korea, so yep, " Duh, Winning!" "Mission Accomplished" and all that jazz.