Probably the main reason is that it is a strategic buffer region.
Also important is that China has a number of regions that are not entirely thrilled with being China. There is a strong feeling that if one region breaks away, it could possibly lead to others, and there is a fairly legitimate fear that this could lead to China’s historical nightmare- the dissolution of China as a whole into a group of independent states. China is not the pillar of strength and unity that it portrays itself. There is a great deal of probably well-founded fear that even a small event could easily break apart the country, which has a strong historical precedent.
Culturally, China is still smarting from the days when it was the center of world, the all-under-heaven. Pretty much every state in east Asia used to send tributes to China. In my opinion, China feels like it has lost it’s rightful place in the world. But there is a sort of feeling that pride can be restored when Taiwan returns to the motherland. But losing land would be a terrible humiliation.
As for everyday people, they get told all kind of stuff. Now and then my students bring up Tibet, and it’s amazing. They were shocked to learn that Tibetans are not universally grateful that China has brought them culture and civilization (and TVs) and brought them out of barbarism. I’ve personally seen TV shows that are all about Tibet being so thankful for being a part of China- people believe this stuff. They believe the Dalai Lama is a bloodthirsty monster, probably a puppet of the CIA, who personally coordinates any violent dissent in Tibet. He gets called a “mass murderer” a lot by my students. When I went to Tibet, my student told me that I would probably get robbed, since Tibetans are armed thieves. They also warned me not to touch anyone, because they believe Tibetans only bathe once a year. In my experience, there is a lot of ignorance of the subject.
Usual disclaimers- I am not a China expert, most of my experiences are from small-town Sichuan, China is a huge country and different people have different views, etc.
I’d also say that a key book to understanding China politically is China, the Fragile Superpower. Written by former Deputy Assistant Secretary of State and UC San Diego professor Susan Shirk, it’s a sober and detailed examination of how nationalism, pride and resentment work on a political level in China. Of the hundred or so books on China I’ve read, this is the most revealing.
This is a…version of the truth. While the Mongols did briefly rule over both Tibet and China, that’s not really equivalent to China ruling Tibet. Some would argue that is like saying Burma is part of India because the British colonized both of them.
In any case, Tibet has a MUCH longer history as an independent state and the events of the 1950s had next to nothing to do with history.