Chip is just a part of the picture, albeit a big part...
If China were to lose the war then US would for sure turn TW into another military base, America's Far East Hawaii of sorts... (not to mention it would be foregone conclusion that by that point the SCS islands would be flipped to US etc)
This means TW island would be Chinese Cuban Missile crisis with hypersonic nuclear tipped weapons ready to decapitate Chinese leadership in under 15 minutes, in effect forcing a full Chinese capitulation (economic, military, technology, political etc) to the US...
On the other hand, if China is successful in reunification, then with TW as China's Eastward Hawaii etc America loses the 1st and 2nd island chain and eventually much of the Western Pacific, and with that loses its hegemonic status along with the lose of the petrodollar hegemony means that America will inevitably implode from within, civil war and getting fractured into many pieces in the homeland....
If ever there were a WWIII scenario, the fight over TW would be it...