Taiwan:
1. PRC has a very clear national security interest in Taiwan not being ruled by an unpredictable regime that may host American weapon systems pointed at the mainland. See Cuban/Soviet Missile Crisis. Coincidentally, similar situation going on with Russia and Ukraine right now.
2. If/when the DPP regime falls and is replaced by a more "independent" government, that government will naturally be favorable to joining the PRC as an autonomous province. Because the economic incentives are there and there is no real ethnic/cultural barrier to reunification (there is already massive brain drain from TW to mainland). This is the outcome that Beijing is clearly planning for and they won't have to fire a single shot to get it.
Uyghurs:
1. Xinjiang is a pathway to Central Asia and critical to the Belt and Road initiative. It's also a relatively undeveloped region that shares a small border with Afghanistan, with a history of widespread poverty and separatism/terrorism.
2. In the mid 2010s, Beijing was busy implementing two important policies in Xinjiang. A deradicalization campaign and an anti-poverty campaign. These are serious, fast-moving policies that involved a lot of heavy-handed action. Like getting millions of people enrolled in vocational training, Mandarin classes, and distant employment. Basically, Beijing wouldn't tolerate Xinjiang becoming Afghanistan 2.0, as much as Washington would love that. So they aggressively developed the region. Ürümqi in 2021 is probably unrecognizable to Ürümqi in 2009, when over a hundred civilians were butchered by ETIM-linked terrorists.
3. In response, Washington, through proxies like Adrian Zenz and WUC, launched a vicious propaganda campaign. Ordinary prisons for violent extremists and other criminals became "concentration camps for Uyghurs". Schools became "reeducation facilities". Vocational training and just having a job became "forced labor". Mandarin lessons became "cultural genocide". Women being lifted out of poverty became "genocide through birthrate suppression". And so on and so forth.
4. The hypocrisy from Washington is mind-blowing. On the "forced labor" front, Washington implemented sanctions against any company that uses "Uyghur labor", in other words, virtually any company that employs Uyghurs. The goal is obvious: to push Uyghurs back into poverty and therefore destabilize the critical region. Because Washington does not care about the lives of distant Muslims; the last three decades make this abundantly clear.