Why would they? England itself is only a minor part of the British Isles. They would however have a chance of being part of a broader Commonwealth of Nations, were it not for a troublesome Declaration they circulated in 1776. This is why claims to "independence" should not be made carelessly, and why official U.S. policy is especially careful to avoid them wrt. Taipei.
You said that Taiwan's language and writing system made them Chinese. By that reasoning, the US would be English. But thanks for clarifying that sharing a language and writing system with another country, doesn't necessarily make you part of that country.
But this is not about "being part of a country"; it's first and foremost about sharing a common culture and history. Facile claims to "Taiwan independence" that ignore these nuances are tantamount to a denial of this shared past and common identity that is part of the continuing lived experience of many people in Taiwan. They are oppressive, dehumanizing and hurt the feelings of the Chinese people who live on the island.
