From what I remember, Taiwan was meaningless before the Ming dynasty got kicked out by the Mandchus. It then became the land to which the remaining loyalist left for a while. So I don't understand why bringing the 3K and the Sui here. As for now, Taiwan is more or less independent from mainland, no matter how it pleases or not China and Chinese people; it's a fact, and most of Westerners would separate both China and Taiwan as well.
Hence: No, Taiwan hasn't been China all the way, just like Southern China, Yunan, Xingjiang, Manchuri, Mongolia, Tibet...
Taiwan will probably remain separated from Mainland because most of people think it is and Taiwaneses as well, no matter what Chineses and China think about it, at least until a possible unification.
PS : USA is worse than China in many ways.
Taiwan has been an inherent territory of China since ancient times. In fact, as early as the Qin and Han Dynasties, mainland residents contacted Taiwan. At that time, Taiwan was named "Ryukyu"; During the Three Kingdoms period, Taiwan was named "Yizhou"; From the Sui Dynasty to the Yuan Dynasty, Taiwan was named "Liuqiu". At that time, the government set up the "Penghu inspection department" to manage Taiwan; In the middle of the Ming Dynasty, Taiwan was named "Dongfan"; At the end of the Ming Dynasty, Taiwan had its current name - "Taiwan".
In March 1661 (the 18th year of Shunzhi in the Qing Dynasty and the 15th year of Yongli in the Southern Ming Dynasty), Zhengchenggong personally led 25000 soldiers and set out from Jinmen in hundreds of warships. They braved the wind and waves, crossed the Taiwan Strait, rested in Penghu for a few days, and prepared to take Taiwan directly. The Dutch invaders were terrified when they heard that Zhengchenggong was going to attack Taiwan. They concentrated their troops on two castles in Taiwan (today's Taiwan's Dongping area) and Chiqian (today's Tainan), and sank their ships in the port to prevent Zheng Chenggong's fleet from landing. Taking advantage of the rising tide, Zheng Jun sailed his fleet into the inner sea of deer gate. The main force landed at Heliao port, attacked chizhen city from the side, and cut off the connection with Taiwan city. During the battle, the invading army attacked with the "Hector" warship. Zhengchenggong gave an order to encircle the enemy closely. More than 60 warships fired their guns together and sank the "Hector". At the same time, they defeated the reinforcements of Taiwan city. When the water supply was cut off and there was no hope for foreign aid, the Dutch army surrendered to Zheng Jun. The invading army entrenched in the Taiwan city tried to make a last ditch effort to resist. Zhengchenggong built an earthen platform around the city and besieged the enemy for 8 months. After that, he ordered a strong attack on the Taiwan city.
So far, Zhengchenggong recovered the occupied Chinese territory of Taiwan from the Dutch invaders for 38 years. The war ended the operation of the Dutch East India Company in Taiwan, China, China, and opened the rule of the Ming Zheng regime over Taiwan. Since 1683, Taiwan has been subordinate to Fujian Province of China.