I started writing out a timeline but I don't know what position you're asking from so I will say for the sake of brevity that the US kept the KMT from being run out of all of China so that the US could us the island as a threat against China -- as it also attempted to do in Korea when it had more-or-less complete control of the southern half. Taiwan spent about 40 years as a military dictatorship killing tens of thousands of dissidents, native Formosans, and others (this was called the "White Terror"), while their patron the US looked the other way while it pumped resources into the country (for the ruling class, mind you) to use the island as a sweatshop site in the interim. This legacy and its connections to fellow US puppet South Korea and US ally Japan go a long way to explaining its current capacity in manufacturing, which make up its other value to the US besides geographical position.

Both Taiwan and SK have made various attempts to assert themselves (with some success in both cases), but with the pathetic diplomatic position of the former and the continued military occupation of the latter by the US, I think "puppet state" is a fair title for them, perhaps as much as Israel, but that's its own can of worms.

I didn't really intend on getting into litigating this topic, but I'm happy to discuss it as best I can.