I think this is the key asymmetry in a US-China war, and whether US aircraft carriers can be sunk or not is irrelevant. At the end of the day, a carrier is a floating air base, and both China and the US need aircraft to provide targeting data for their munitions. In a sense Chinese airstrips are even more vulnerable to missile strikes as they can't move or hide, yet they can't be attacked because that could invite nuclear escalation. If sinking a US aircraft carrier posed the same risk of nuclear escalation would we even be worrying about their vulnerability to missile strikes?
This seems more like a strategic problem than a tactical problem regarding carrier defense. If China is more committed and willing to escalate than the US, then sooner or later there'll be a line where China's willing to cross but the US won't.