It's important to note as well that Chinese intervention isn't a binary, fully-on or fully-off kind of thing; it's possible to imagine various degrees of intervention.
One that seems likely to me in this kind of scenario, for instance, would be for Chinese forces to move in to uphold the integrity of the pre-war border. Allied forces would be free to push the North Koreans out of South Korean territory; but if they tried to push beyond that towards Korean reunification or some kind of territorial gain at the North's expense, they'd have to fight Chinese troops to do it. This would let the Chinese get involved but still pose as a neutral arbitrator rather than a direct combatant or contributor to escalation, since their nominal goal would be resumption of the status quo ante bellum.