Few people in the West believe what the Chinese say about him. Simple as that. But because of political pressure from China, the governments here don't feel able to say so openly, because China is so powerful, especially economically.
President Obama met him in a personal but not in a political capacity. For that matter, the Dalai Lama has now resigned his political position and is no longer leader, and has no political position at all in the government in exile.
So this makes it easier for him to interact with others in the West. He does so now as a spiritual teacher in exile from his country of birth - rather than as a political leader.
Incidentally he has never been a spiritual leader of Tibet in the sense of the Pope. The Buddha made it clear in his sutras that nobody could take on the role of leader of the community after he died - but that we should take the teachings themselves as our leader - and that we should examine them also for ourselves and never say that something has to be true because the Buddha said it.
So - the Dalai Lama is not in a position to tell any other Buddhist what they should believe or how they should act, in Tibet or outside of Tibet, and never has had that authority. He can present teachings expounding on his understanding of the Buddhist teachings. But it has always been understood by Buddhist listening to him that he has no authority at all to tell them what to think or do on anything. A different situation from the Pope who can issue papal edicts.
As an example - there are various awkwardnesses in the statement of the Buddhist monks vows, because they were composed in a different society from our present day world. And even more so in the case of the nun's vows. But no Buddhist anywhere has the authority to change them. Though - there is a flexibility involved, that the monks and nuns have to decide how to interpret them - so other Buddhist teachers can help by offering suggestions about how to interpret the vows in the context of the modern world, and how to apply them. But again nobody has any authority there to say "this is how they should be interpreted" either.
It is just personal advice, teacher to student, to someone who goes to them for advice. And another teacher might well offer differing advice. That's just how it works, for all the teachings.