This isn't the whole truth I don't think. They do maintain officially maintain this line... under the implied threat of invasion/war from the mainland if they deviate. Similar to how he US officially doesn't recognize Taiwan as a country because it isn't in the US interests to invoke hell's fury from the PRC over Taiwan, but for all intents and purposes the US treats Taiwan as an independent state.
The DPP (plurality party in Taiwan) generally leans towards viewing Taiwan as an independent nation. The main opposing party's view is generally consistent with what you've stated in my understanding.
It's a complicated topic but you're mainly projecting the main land's view onto the people of Taiwan.
Take a stroll by these.
Another note: History is written by the victors.
Depends on the view. The central Chinese government certainly considers Taiwan to be its province, and achieved that Taiwan has no seat in the UN, and sanctions Hotel chains that treat Taiwan as separate country. Taiwan themselves maintain a claim to the entire mainland as well, they just disagree on who should hold power. There are also countries that recognize Taiwanese independence, as there are countries that recognize Taiwan's claim over of all of China.