But yeah, it's hard to keep it separate. You want to support the people of Tibet, the people of Tibet seem to support the Dalai Lama... not monolithically but it's possible most do... and of course the Chinese invasion and occupation increases support of the Lama among Tibet people, for unity against the incursion, etc.
In fact, if China hadn't invaded and occupied Tibet and turned the Dalai Lama into a symbol, then it's reasonable to think it would be much more likely there would be more internal opposition to the Dalai Lama. If it had existed as an independent country all these decades, would it still be that kind of theocracy in 2023? Seems unlikely. The invasion froze Tibetan internal politics.
We see this dynamic of "falling in line" against an obvious oppressor all over the place. Like, try opposing Putin's Russian invasion of Ukraine and thinking there are fascists at the core of the Azov brigade and being wary of US/NATO power too. It doesn't make you many friends, you just have everyone thinking you're a traitor on all sides. It's way easier to just pick a side, from what you are told are the two sides to pick from, that everyone else is picking from.
AND we see the dynamic all over the place -- invasion and occupation by a foreign entity which exerszies power on ethnic/racial/national grounds... generally leads to more support for local despots than there would be otherwise. The internal conflicts and power struggles and movements for freedom are put aside, as everyone joins in unity to resist the external invader OR as internal politics are just scrambled and destabilized. You never get more democracy from an invasion. (Sorry, Iraq).
That'd be an issue if the support was more real. As it is the "show support" requirement is just that you do the "show" part. Usually any effective action is ignored or even frowned on.
It's not about the people being "supported" its about the your neighbors who see your bumper sticker.
Isn't "Free Tibet" kind of passe? Like it was big in the 80s or 90s, but it's not a trendy issue anymore.
But in any case, the attitude that you expect progressives to have is one that leads to paralysis and inactivity. Nothing is ever going to perfectly conform to one's personal or factional beliefs, so if perfect conformance is a requirement you can't support anything expect yourself. Realistically, once has to reduce something to its parts and then ignore some, or pick the best of "bad" options. In the case of Tibet, (more sensibly) that's a desire to support national self-determination in the face of foreign occupation (by authoritarian Communists, no less) before all other issues, (less sensibly) that's ignorance that lets a progressive project his own attitudes and beliefs on some foreign group and not notice the error.
Another, more jarring, example is Palestine. If Israel ceased to exist and the Palestinians got their own state, I'd expect that new state to be close to the top of the progressives' shit list. After all, conservative Muslims aren't very friendly to LGBT stuff.