The Cold war is a collection of proxy wars, one of which is Korean war. Did China win the Korean war? No, it's a stalemate. Maybe if China wins Korean war, then Korea will abandon it's alliance. I don't see a situation where being big economy or population will result in voluntary US expulsion from peninsula, or voluntary Korea ending the alliance. It takes winning the Korean war to do that. (And also no, I'm not expecting the US collapsing like Soviet Union, although that would be wonderful...even in that scenario of US collapse, what makes you think SK won't go nuclear immediately?)
The South Koreans have zero spine to stand up to Americans, they have no independent foreign policy. The status quo should be permanent.
Cold war was between US and USSR. Not every conflict involving one of the two should be lumped in cold war. If you involves everyone (NK and China), then it is not finished, then why would people talk about cold war 2.0? So don't put everything in a big mess, otherwise everything lost meaning.
For NK and US, it was a stalemate as both of them started and stopped at the same spot. China entered the war at the time when US troops were standing right at the south bank of Yalu river. China ended the war standing right at the 38 parallel line 200km south. Now you tell us what is a win.
China's objective in that war was the 38 line, no more no less. That was why China openly warned the US not to cross it, and China only entered the war when US crossed that line. China made it clear via Indian ambassador that the red line is for the US and its proxy troops, not SK.
The Koreans certainly have a big say to the future of the peninsular. China as its direct neighbour certainly has a big say. But China is not US and will never be. You can rest your concern, China has lasted for thousands of years without acting like the west, and we are outlasting them.