"Different language, ethnicity and culture, their own government. What happened to self-determination? Isn't it Tibetans who get to decide if they're part of China?"
Yes, but Tibetans have pointedly avoided pressing the demand for national independence in favour of autonomy. That has been the Dalai Lama's demand, and I think it is reasonable to assume that he reflects mainstream ethnic Tibetan attitudes. There were at various times currents that pushed rhetorically for independence, but they have always had to deal with the basic problem that the monastic leadership of Tibet is more afraid of popular mobilisation than it is of Chinese domination, and so has always pulled back from a straight-forward demand for independence.
The main proponents of Tibetan independence were not Tibetans, but the CIA officers who trained Tibetan guerillas in Camp Hale in Colorado in the 1950s and beyond. But they, as the Dalai Lama said, were only using the demand of independence to put pressure on the Chinese, and when policy changed under Nixon, the plug was pulled on the whole program. The CIA did help set up Tibet House, which remains a powerful centre of anti-Chinese propaganda to this day.
Could the radical wing of Tibetan ethnic identity pull itself away from the Dalai Lama and the religious leadership? I do not think so, since the monasteries are the repository of what cultural identity there is.
Freedom for greater Tibet would be politically undemocratic, since Tibetans are a minority there, and would rule over a Han majority. Could you make a free state out of the current Tibetan Autonomous Region? I do not think so. If it was free, it would not be Tibet: the irreducible element of Tibetan ethnicity is subservience to religious authority, indifference to secular authority.
Certainly the Chinese authorities have done enough to show that they do not have the moral authority to run Tibet, but sadly the Tibetans have not sought their own country, only the preservation of the Dalai Lama.