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.
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.
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.
While first strike sounds good, don't forget that there are nuclear subs everywhere in the world, at all times, just waiting for the word.
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.
For some reason, most Americans seem to discount the fact that weapons are a two-way street.