I'll be more precise -- the Qing dynasty (Manchurian) emperor that asserted state power over theocratic power in Tibet was himself personally a Buddhist. This was from a paper I read -- it's a book review of Forging the Golden Urn ; the book is written by Max Oidtmann, and the review was written by Matthew Mosca, published in the Journal of Chinese Studies. In the review, it talked about the Qing emperor trying to assert state power without alienating the Hans. I don't know if all of Qing were Buddhist.

I am also not saying all of the ethnic Manchurians were Buddhists. I didn't get that deeply into the history.

I do know that the Mongolians greatly respected the tulkas of Tibet, with branches of the Tibetan Buddhis monastaries found in Mongolia. The search for reincarnations also extends out into Mongolia. The review mentioned that -- during the Song dynasty, the Mongolians let the Tibetans have enough autonomy to concern the Hans, and politically, they were seeing if the Qing emperors will do the same. The creation of the Golden Urn lottery system was a way to assert state power while also respecting the religious process in Tibet. (Whereas, in contrast, I think the Chinese selection of the Pachan Lama and control of the "tulka" was done as part of the cultural assimilation and oppression program by the Chinese). But yeah, the impetus for the Chinese conquest of Tibet in the twentieth century was centuries in the making.