I'm pretty sure the bill came out long before the Chinese exercises so the bill is not about "saving face" it's about policy. Since international law doesn't support interference in Chinese internal affairs, the US passes local laws to legitimise it's otherwise illegitimate actions. The US always does that for example that's what they did with Meng Wangzhou, by international standards they basically abducted her yet they would defend what they did to her by passing local laws and saying "hey we are just following the law, why are those Chinese against following the law".
Defend Taiwan? You are talking as if I was saying the US will declare war on China. and any way in case of a war they won't station ships in Taiwan, I specifically said the US has no intention of waging war on China. Officially the US already has a small number of soldiers in Taiwan, unofficially the number is probably higher. To de-fang China doesn't mean the US will launch a preemptive strike on the mainland from Taiwan, it just means to have a certain military advantage over China to make it less bold in it's foreign policy.
Taiwan is just a pawn, the US will never go to war with China over Taiwan no matter what, well unless if the US is ruled by an absolute hawkish right winger in which case yeah may be they would defend Taiwan who knows.
In regards to "public outcry", the only outcry in the US is the outcry that is hostile to China, most Americans actually want to go to war with China.
"When asked about a range of potential scenarios, just
over half of Americans (52%) favor using US troops to defend if China were to invade the island."