The island was pretty far away, and the later phases of the empire were not conducive to seafaring. As far as controlling it, the Chinese lost it fairly easily to a Portuguese fleet…later, the JJapanese took it. Taiwan had a farily large aboriginal population, until the Nationalists decamped to the island in 1948. So was it of much importance to Imperial China?
Short answer is “no”. Taiwan was never strategic to China historically.
The history of China for several millennia - basically as the empire got weak and fell into chaos, the first sign was that the peripheral provinces and tributary kingdoms declared their independence. With strengthened central government, they retook these areas. This pattern continued with Formosa, for example, in 1947. They also saw the pattern with Shanghai international areas in the late 1800’s.
So the current Chinese government cannot graciously accept political reality any more than Abraham Lincoln could with the south - this is why they keep denying that Taiwan is a separate country, or Tibet (in and out of tributary state / province status over the centuries), it’s why the Uighurs in the west are considered heretical traitorous terrorists, and why they can’t even accept a bunch of islands in the ocean - to the east or to the south - might be given away by treaty. To ceded territory would be to suggest the government is weakening and on a downward spiral. They have also repatriated Hong Kong and Macau despite the difficulties in absorbing previously independent populations.
Although China is one of the larger land masses in the world, they are always out for more arable land with drinkable water and minerals that can be mined and sold. They didn’t “historically need” the Spratly Islands either, but will likely use their considerable navy to enforce their land claims there. They don’t make a huge distinction between what was theirs 150 years ago and what they claim today.
Edited to add that the Chinese don’t make a huge distinction between what they were a part of 1,500 years ago and today. Ie, Anything that was part of the Mongolian empire (circa 1271 - 1368) is ipso facto a part of China.
Taiwan was just a pimple on the ass of the empire. The island was sparsely populated by Fukienese and Hakka, as well as the aboriginal Polynesians. It wasn’t fully controlled by Qing China. The Japanese sorta took over in 1895 post Treaty of Shimonoseki and solidified control by 1915. Here’s wiki’s take.