the only claim being appeased is to what they already control, Taiwan

That's not true, or at least what I would argue. You can point me to any article where some Western politician is saying "as long as Taiwan want it's island we support that, but not more than that". In fact, I don't know of any conditions the US or anybody who defends Taiwanese independence, is making regarding their claims. There is no "Taiwan only" constitution that the US supports. This is the needle in the ass of the PRC. I think it would be a different situation, if Taiwan (and the US) would say "we want Taiwan to be its own country, and we recognize the PRC as the successor of China.

But they don't do that. They actually support the ROC and everything on their constitution. Including the 11-dash line in the South China Sea, that is larger than what China is drawing with their 9-dash line That they are for the "will of the Taiwanese to just be independant on their island" is for the public of the G7 countries. Nobody is willing to give up the territories of ROC afaik. Yes the ROC can't do anything about it in terms of military power, but they equally don't make any steps to remove them. (But I think if the US tells it's guys at the DPP to create such a constitution that claims only the island of Taiwan, they will only do it to provocate an attack by China. But that's beyond my point and the map above.)