Taiwan may be a core objective of China, but Taiwan is more than a question of ownership of that island. We can see that in Chinese cyber-war:
Taiwan has been under heavy attack by Chinese hackers for over a decade. But government cyber security experts, while meeting and comparing notes with their American counterparts have determined that Taiwan appears to have been where Chinese practice and try out new cyber weapons before using them on the United States and the rest of the world.
So China doesn't wage cyber-war on Taiwan for the sake of defeating Taiwan, but to be ready for the world battlefield.
As China builds up their military power and pushes neighbors around to reestablish China's tradition of doing what they want in the area, they find American power in the way.
We both stand directly in their way and bolster smaller nations near China who otherwise would know their place and shut the eff up when China tells them what to do.
So as we pivot to the Pacific (whether it really puts more power in the region as our overall force structure declines is another question), we encourage regional powers to resist China and make it more likely that we will resist China.
Obviously, if China is to get their way in "their" region, they need to knock us down a peg. That doesn't mean China has to defeat us. It means China has to defeat one of our allies in a way that demonstrates Chinese mastery of local wars under high-tech conditions.
If China can invade and defeat the Taiwanese before we can get our act together, mobilize our superior forces, and effectively intervene with our superior military power, China will demonstrate that American security guarantees will not stop China.
Sure, nations may adapt to this new Chinese capability to mass decisive power at the point of conflict close to China for long enough to win a campaign (Japan and South Korea, for example, might go nuclear), but we will have been visibly beaten if Taiwan falls. And some nations will decide to bend to China's will rather than risk having their independence snuffed out.
So rather than being a cause of US-Chinese friction (remember we worked closely against the USSR despite our support of Taiwan), Taiwan is an asset that if crushed by the Chinese military, China will exploit diplomatically to break down our alliances in the region, and practically to both demonstrate their arrival on the world stage as a military power and to open the path to project power further into the Pacific from a Taiwan base:
Why would we think that letting China do what they want to Taiwan would improve Chinese relations with us and reduce the chance of war when gaining Taiwan just puts China in the position to project power further out to sea? Wouldn't we just be faced with the same problem further south and east but without Taiwan as an asset on our side penning Chinese naval forces close to China?
The status of Taiwan is not just a Taiwan question. Conquering the island democracy would be a signal victory heralding China's rise and America's decline in the western Pacific.
