China wants direct control over the chip fabrication capabilities of TSMC, which produces ~90% of microchip fabrication in the 5-10nm range and ~60% of all microchip fabrication. Exerting control over the rest of the world's access to advanced microchip fabrication is the primary goal. Don't pretend China's aggressive behavior is about anything else.

Taiwan and China are tightly integrated in terms of culture (the best selling artist in China is Taiwanese, for example), and bilateral migration between China and Taiwan is extremely high.

Less than 12% of Taiwanese citizens support unification with the PRC, while 50% support Taiwanese independence and 25% support maintaining the status quo (see section 10). Additionally, 62% support Taiwan seeking international recognition as a sovereign nation (section 6).

Meanwhile, Taiwan is literally a fortress with a massive force of military-trained personnel.

Of course it is, they are being threatened by an aggressive authoritarian nation with a vastly larger military.

"Approval ratings" are rather nebulous. By the divisive and partisan nature of American politics, approval ratings in America are naturally going to be low because both parties exist solely to shit on each other. In China, "approval ratings" get measured from the perspective of "is my life improving?" rather than "would my life be improving more under someone else?"

Honestly? I think asking if someone's life has improved is a more fair polling question to ask, but it's one that's difficult to differentiate in the US because of how radicalized everyone is.

Basically, what I'm saying is that the US would have a higher effective approval rating in the Chinese context than it does today, because many American lives ARE improving under the American government. People just think (often incorrectly) that it would improve more if the other party had power.