Why? I actually think nukes would be fairly unlikely. The American strategic nuclear arsenal is capable of utterly destroying China, while the Chinese one is only capable of hurting the USA. (Badly, yes, but there would be a viable state left over.)

This means that the Chinese will be very unlikely to want to escalate to a nuclear conflict, while for the US the damage that the Chinese can inflict is also a sufficient deterrent. This is reflected in the published policies of both countries: China has a strict, publicized no-first-use pledge. US frequently flirts with the concept and is generally only prevented from adopting it by NATO allies.

I think you've identified the reason: the US war planners see advantage to some "limited" nuclear exchange and are already flirting with the idea of it (not sure what gives them the confidence that it will stay within their limits if they escalate to nukes). The temptation is going to be irresistible if things start looking bad for them (and I think in a more conventional war China enjoys an advantage here purely from geography).

Even from the Chinese perspective, once they're in an active war that threatens their basic territorial integrity, who knows if they can really stick to NFU.