[waterheater]

Other folks have highlighted the basics, so here's an analogous situation to think about: imagine that the US civil war saw the CSA taking over the territory of the USA, with the US federal government moving to a large hypothetical island off the East Coast. The CSA never completely eliminated all constituent US elements, so the USA still technically exists.

Fast forward to sixty years after the fighting just stopped one day. How would you view those two political entities? Are they equal? Does the existence of the CSA mean the USA doesn't exist as a country? What if the CSA implements trade policies which dissuade you from recognizing the USA as an independent nation? Does the USA still exist in your eyes? Is the USA or the CSA the "true" country?

The answer at this point in time is quite simple: the Republic of China (aka Taiwan) is clearly an independent country. However, the People's Republic of China (aka China) holds such economic power, they bully the rest of the world (though trade policies, etc.) into not recognizing Taiwan as an independent country. Also, the combination of geography and maritime law means that if Taiwan is an independent country, China has very little direct access to the open seas. They want freedom of navigation, so there's a geopolitical angle as well. Look into the construction of artificial islands in the South China Sea to advance the Nine Dashed Line policy.

You also need to remember that Chinese civilization is extremely old. The PRC claims to hold something called the Mandate of Heaven, which is a fairly important concept in Chinese history, as it is divine authority to rule over the Chinese people. Frankly, based on how the PRC acts, the RC clearly holds the Mandate of Heaven these days.

[samus]

China would have open access to the oceans even without controlling Taiwan. The same rules that deny China territorial rights to the South China Sea also grant it access to the oceans. The Nine Dash line is about territory. China wants the South China Sea to be considered its territorial waters where it can make stronger demands about who passes through and what they could do there. It's not the only country trying to do something like that, but international law generally only recognizes the sea up to 12 miles out to be territorial waters.

The Mandate of Heaven is rather more like a justification of de-facto rule and claiming to have lost it is an argument against a failing dynasty. The dynasty managing to unite China would declare it, and as it eventually collapses, rebels would declare it to be forfeit. There would be messy civil war, one of the warring groups would dominate all the others, and the cycle starts anew.

[sandspar]

The Taiwan Strait is extremely efficient for shipping routes. The sea and wind to the East of Taiwan are treacherous. Going the long way round would be very difficult and greatly increase shipping costs for a substantial portion of the world's goods.

[sandspar]

Taiwan is not clearly independent. The issue was treated in Treaty of San Francisco, yet the treaty never unambiguously declared whether Taiwan is independent. It's an extremely complex legal issue. You can state your case but understand that the opposing case has just as much legal ground.