> Why does China even care about Taiwan? <
I would hazard two guesses:
1. It’s a national pride thing. And the Chinese are very big on national pride. As an analogy, suppose the Confederates still held Florida after our Civil War ended. The Federal government would resent that. Now for China/Taiwan, multiply that resentment by 10.
2. Things are looking a bit rough for China going forward. And wars are always great distractions.
Taiwan would give China control of the eastern pacific.