> I would hope that China would understand Buddhism enough to know that there is no possible separate authority... I mean, the Heart sutra almost definitely came from China..
Speaking as an agnostic, outside observer looking in on the various forms of Buddhism:
Buddhism has mutated a lot in its history, and every country it touches has been changed by it and in turn changes it. I don't think any two countries practice the same form of Buddhism. Some sects of it are strongly mystical and others are more rational, and deification is an important part of some practices (and practitioners) and not others. In the US, I've been to supposedly Zen practices that had an emphasis on "acknowledging/honoring" the bodhisattvas prior to meditation; to me, as an agnostic, it felt very similar to worship.
"Buddhism with Chinese characteristics" (to steal a phrase) mixes their particular cultural norms with the Indian version, adds on Confucianism and Taoism to some degree, and then seeks to put it all under CCP control. Much like the Romans absorbed Christianity and added their own flavor, China does the same with its assimilees... again, for cultural and political hegemony. They are trying to weave the young puppet Dalai Lama into their own narrative, slowly and subtly.
You are a Buddhist in the same sense that Protestants and Catholics are both Christian, but that never stopped them from disagreeing :) Weird, the arbitrary lines in the sand we like to draw, no?