Korea was a proxy war between the US and China (i.e. Communism). The US wanted a foothold on the mainland and to put a demonstration of the virtues of Capitalist Economics in the Communists' back yard, both of which it got in the form of South Korea. If the US hadn't been involved, "Democratic People's Republic of Korea" (i.e. North Korea) would have rolled south with China's backing and "Korea" would have been a single communist country.

Without China's involvement the opposite would have happened. China was well aware of this and did not want a capitalist economy sharing a border with it -- Berlin Wall kind of problems -- so they weren't going to stop supporting North Korea.

Which is the kind of situation that makes unconditional military victory close to impossible. China just keeps giving North Korea assistance, most of the casualties are North Koreans rather than Chinese and the North Korean leadership won't surrender as long as they have China's backing because it's not a democracy so staying in power is the only thing their leaders care about. "Unconditional victory" would effectively have to be a military victory over China.

At the time the US had nuclear weapons and China didn't, but China had half a billion in population even then, and they too were a non-democracy whose leaders cared more about staying in power than dead citizens. The US could have won militarily if they were willing to kill several hundred million people. It's probably best that they weren't.

And it was largely the same for the other proxy wars. Unconditional victory would require victory over not just the nominal enemy but the ones propping them up. Hence Cold War.