Would the US be willing to launch a bunch of non-nuclear missiles to China, though? Because I'd also bet they wouldn't. And if they would be, the missiles they already have would already work
This is all wild speculation on our part, as AFAIK, there hasn't been any direct attacks between two nations that have nuclear bombs except some skirmishes here or there (and being able to do that doesn't warrant an extra $24B spending). At most, war by satellite (USA supports country A, Russia supports country B, and A and B bomb each other). Maybe I'm being unimaginative, but I can't see how direct attacks between China and US wouldn't escalate to a nuclear war, even just due to egos.
When the US and Soviet Union established MAD the prevailing method of fighting wars was to fire/drop/shoot as many low-accuracy munitions as possible at the enemy. More ordinance was expended in the Korean war than in WW2, and by the Vietnam war planners had to moderate B-52 carpet bombing as 12x B-52s carry nearly a nukes worth of ordinance. North Korea lost roughly ~20% of its population in a conventional war.
It's easy to justify nuclear escalation when fighting this form of total war, and difficult to when the only casualties are military and the bombs mostly fall on equipment. While a land invasion or surprise strike of the continental US/EU/China/Russia would surely trigger a nuclear response, a direct battle over territories/international policy wouldn't be as clear cut.
Not to China, but to Chinese forces (even those within China controlled areas), I expect they would as does China or they would have already moved. China may ultimately win an exchange of value, but it ultimately is more exposed to other threats like Japan, Russia, India and internal struggles. China can't overcommit and win a battle to lose a war.
China would launch their own defensive missiles, as well as offensive missiles at aircraft carriers in the South China Sea and nearby US landing strips. This is exactly the wargames that they are discussing that the US continually loses at because they can't defend their base infrastructure in those areas.
If you state a MAD policy, how can your opponents be sure that the bunch of missiles you're sending aren't nuclear? They need to make a decision before those missiles detonate or else their own nuclear capability might be removed by the first salvo.
Sure, but my point is that, from China's perspective, they don't have the opportunity to validate that the swarm of missiles coming their way aren't nuclear. They have to assume that they are or they risk losing a large amount of their retaliation capabilities.
So to answer the question
> Would the US be willing to launch a bunch of non-nuclear missiles to China
No, because China has to treat them as nuclear either way.
If China began to bomb Taiwan, the US absolutely should launch non-nuclear missiles at Chinese bombers / fighters / drones / missiles.
MAD assures that China won't use nuclear, and the US won't use nuclear either. But normal non-nuclear combat is fair game.