Historically the Korean war was a UN victory. South Korea's sovereignty was preserved (plus the addition of some North Korean territory), the North Korean invasion totally defeated, the Chinese attempt to push the UN out of Korea also repulsed, and the Communist side sued for peace.
Having the UN side totally conquer North Korea would require bombing the Yalu river bridges and a greater commitment of forces. If the UN can muscle their way up to the Yalu they can simply hold indefinitely against any attempted Chinese crossings.