Since the death of Chairman Mao, beyond a few moments of tension, the governments of the People's Republic of China (PRC) and the Republic of China (Taiwan) (ROC) have, until recently, by and large, accepted the status quo with some degree of comfort. I'll use their official names because it matters, and because the efforts to control the language emanate from Beijing, and I'd rather treat the two governments the way competing governments for territory have been treated elsewhere - as two sovereign states.
You see the Korean Peninsula has two governments on it, both claim de-jure legitimacy over the entire territory of Korea. The Republic of Korea (ROK) in Seoul and the Democratic People's Republic of Korea (DPRK) in Pyongyang, both being rivals who have fought to the death to conquer and destroy the other. The DPRK to this day displays its maps including the whole peninsula, as it claims the south is illegally occupied by the United States which governs the south under its "fascist puppet-clique".
China has the same, albeit on a much less balanced scale. The ROK and DPRK both govern roughly similar sizes of territory, if not populations and not remotely economies. The PRC on the other hand is a giant against the minnow of the ROC. Like its ideological comrades in Pyongyang, the PRC has never treated the territory of the ROC (Taiwan) as anything other than renegades that need reunification into the motherland. The ROC on the other hand is barely the ROC at all. Brave but unrealistic rhetoric of "retaking the mainland" faded away after the semi-fascist President Chiang Kai-Shek died in 1975, given the US stopped seriously thinking about assisting in that endeavour after the early 1950s. Very few in Taiwan now think that the ROC will once again fly its flag over Nanjing and replace the PRC, what they really want is to be left in peace, which is only reasonable.
ROC (Taiwan) today is a sovereign state that is in so many ways unlike the stultifying military state of Chiang Kai-Shek, being a vibrant liberal democracy, open economy and basic freedoms that are absent under the PRC. Whether or not people in Taiwan want to declare formal independence and unshackle themselves from the legacy of the ROC wouldn't change how they manage their own affairs, except for one shadow overhanging them - the PRC's demand that they succumb to its control.
Xi Jinping has clearly made a play, far beyond the rhetoric of previous PRC leaders, to take military steps to take (not retake, as the communists never had it) Taiwan and the islets under its control. Let's be clear, the PRC never revoked its "right" to do so, but that had largely been rhetorical. Note also that from the late 1990s through the 2000s relations between the PRC and the ROC improved significantly, with direct trade and investment allowed, along with direct mail, flights and travel permitted. The pre-Xi strategy regarding Taiwan was to treat it like "just another Chinese province" in the hope that this would make people of Taiwan feel more relaxed about a future with the PRC, but also interlink their economies so the economic dependency of Taiwan would shift from exporting to the world, to trading with the mainland.
Xi has now blown all of this up in his quest to take a more aggressive stance. What is Xi's motivation? Is it to distract from problems at home? Is it to claim glory that even Mao could not succeed? Is it to demonstrate that the PRC of the 21st century can be the leading power in the Western Pacific and show its neighbours that the USA is no longer in charge? Probably all of this, and given this it is absolutely essential that the USA, with its allies, make it clear to Xi that none of this will be accepted.
It should be crystal clear to Beijing that any military aggression towards ROC forces, or territory, will be met with a military response in support of the government in Taipei. Why? Because the international order depends on it, and because the consequences of the alternative are likely to be much more destructive to international peace and security, than taking on Beijing on a limited scale.
1. International rule of law and the value of peace: The PRC can proclaim endlessly that Taiwan is an "internal affair", and there is no shortage of proclamations from Beijing about the principle of "non-interference in each others' internal affairs" (a principle Beijing has broken more than once in many countries), but fundamentally ROC (Taiwan) is de-facto and de-jure an independent sovereign state. It has all of the characteristics of a sovereign state, and only lack extensive international formal diplomatic recognition SOLELY because the PRC prohibits it. The Cold War saw most countries recognise the PRC when the PRC and ROC were in diplomatic competition, because the PRC was more useful as a bulwark against the USSR. Since the end of the Cold War, the growth in the PRC economy has meant this has continued to be more important than ROC (Taiwan). The ROC dropped the insistence of choosing between Beijing and Taipei for diplomatic relations, but the PRC continues to bully the international community into the legal fiction that the ROC (Taiwan) isn't a sovereign state and has no rights that Beijing does not "let" it have. So from that, the idea that any government can use military force to address its dispute with another government is unacceptable under international law and the UN Charter.
Furthermore, it doesn't even matter if the PRC treats the ROC (Taiwan) as its own, it does not justify military force to win an internal dispute. The reality that Beijing knows is that the ROC (Taiwan) is independent, because it acts independently. Countries cannot be allowed to just declare that they have the right to inflict force on another sovereign entity to take its territory. It's a norm of international law, and if the PRC wants to be a respected member of the international community, the latter must make it clear than attacking ROC (Taiwan) territory is an unacceptable risk to international peace and security (even though Beijing would veto any action in the UN Security Council. The message must be loud and clear that no power can just exercise force to take territory and try to overthrow government that it doesn't like. Russia did the former with barely a blink internationally, under the Obama era, Biden cannot let this happen to Taiwan.
2. Dissuading further aggression: If the PRC can "get away" with taking parts of the territory under the control of ROC (Taiwan) or go further, then it will embolden it to go further. It has territorial claims of India and Pakistan, and of course its ongoing active claims in the South China Sea. There is little sense that Beijing is expansionist in the way the USSR was, but the willingness to use military force (and the likely casualties from that) if not resisted by forces outside Taiwan will generate new interest in settling scores or supporting those that may want to settle scores. As crazy as it is, this might include the DPRK seeing Beijing as willing to back it, once more, in an attack on the ROK. Intervention is about deterring more military force. If Beijing takes Taiwan, there are the Senkaku (Diaoyu) Islands just to its north-east, controlled by Japan, that are claimed by Beijing (and in fact Taipei too). Expect them to be under pressure, and if the US wouldn't defend Taiwan, would it really defend the Senkaku Islands? If not, how vulnerable would Japan feel and how abandoned by the US?
3. Pax Americana remains intact: Since 1945 the Pacific has been dominated by the peace established by US victory in WW2. The USSR had limited impact on that, but the PRC wants to remould the international order to suit its needs, one of those is to expand its sphere of influence and to demonstrate that it is the leading naval and air power in East Asia. If the US demonstrates that its commitment to Taiwan's defence is toothless, then it will be a transformative end to Pax Americana. The implications of that aren't just in the US feeling shame and impotent, but the reaction of other governments in Asia. The ROK, Japan, the Philippines, Indonesia, Malaysia, India, Vietnam will all wonder if they are safe in a Pax Sinica (Marxist-Leninist). Given the PRC has WMDs, there would be some fear in Seoul, Tokyo and Delhi that the US's nuclear umbrella has just folded up somewhat, and more disconcertingly they should all develop or expand their own. ANY of this will aggravate the international order in Asia, as a nuclear armed ROK and expanded nuclear India will raise temperatures and risk. Japan will be unlikely to embrace nuclear weapons, but it will want to expand militarily to deter the risk the PRC will take islands near Taiwan that are under dispute, next.
So if you worry less about Taiwan, worry more about what is next, Asian powers seeking to contain the PRC with WMDs. Beyond that must be the effects this has on US power elsewhere, such as in Europe.
4. Liberal democracies must be protected from tyrannies: It is almost least important, but morally it is easily the point that pulls the hardest at emotions. ROC (Taiwan) is a vibrant, liberal democratic capitalist country of free people. It demonstrates that China can be a liberal democracy, and that Beijing's long held excuse that China "cannot become a democracy" because it would be "ungovernable". ROC (Taiwan) has, like the ROK, transformed itself from essentially a dictatorship into a free and thriving liberal democracy. It should not be able to have that destroyed by the world's deadliest dictatorship (60 million Chinese killed through famine and executions in the Great Leap Forward and Cultural Revolution). ROC (Taiwan) has been a model international citizen, it has been peaceful, productive and constructive internationally, it represents values that the US should hold dear, and which promote peace and development globally. If Beijing has snuff that out with only some stern words from Washington, then Washington will have abandoned any pretence about leading by some moral imperative.
So there should be clear solidarity with ROC (Taiwan). It should be no doubt at all that if the PRC attacks military forces in Taiwan or any territory, that there will be intervention to counter this, and that unilateral sanctions will be imposed on the PRC by liberal democracies. The UN is toothless on this, not least because the PRC has a veto in the UN Security Council.
There are lots of reasons to be critical, historically, of Pax Americana after 1945, with a good list of mistakes made, and behaviour that is not becoming of a country that proclaims itself to be in favour of freedom. However, there is little doubt that a world led by a flawed semi-free liberal democracy is better than one led by a Marxist-Leninist one-party state.
(Oh and don't doubt that the Socialist Republic of Vietnam is much more on the side of the USA than the PRC)