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.
China or whomever attacking a carrier to destroy it with anything other than 1945 style surface warfare tactics leads to nuclear war, even if it's not the final decision point. The response to such an attack leads to nuclear war in enough projections.
In the same way, the US should keep strikes on Chinese military facilities on the table. But escalation is much likelier because those facilities are located in civilian areas and near nuclear missile sites. That's a perfect analogue of the strategic asymmetry. China and it's civilian population actually lives in the area. The US is just visiting with military assets, and losing a civilian is a tragedy but losing a member of the military is the job they signed up for.
US wargaming had repeatedly found that escalation to nuclear exchanges is all but a given if serious attempts are made by either side to destroy naval and air base capability, and it's safe to assume their PLA counterparts reached similar conclusions.
As an example, in a Chinese move against Taiwan or Japan, anything that crossed into attack range would get destroyed, but there would be no counter-attacks against Chinese facilities and bases. Does that play out in reality? Let's hope we don't find out.