You're really bad at googling, then, as you should have hit upon a basic description of Mosuo political structure within the first few pages the search engine suggested to you. All meaninglful authority in Mosuo society outside of the official Chinese govenrment is family based, and the head of the family is its eldest female member, called the Ah mi of her clan. All economic, social, and political control ultimately come back to her "office", symbolized by her ownership of a private room and the keys to the family larder, passed down matrilineally.
Your obfuscation about matrilineality is a red herring, as a society easily be both matrilineal and matriarchal. While not all matrilineal societies are matriarchal, some are. For that matter, patriarchal and matriarchal systems of authority can co-exist in the same society as well, though one usually ends up taking precedence over the other, almost always but not always the partriarchal system.
Your "general rule in anthropology" is fictional, there is no such general rule in anthropology. In anthropology, as an actual general rule, we derive terminology from descriptions of observed societies, rather than trying to impose our cultural assumptions on to them. I can't imagine a professional anthropologist finding such a rule useful even if it were true. There being so few well-known matriarchal societies, what woud be a credible data set to derive such a rule from, and who would we apply it to? A prescriptive rule never takes precedence over observation, and the known matriarchal societies are all well documented, so why would you rely on a "rule of thumb" rather than doing ten seconds of googling and learning the correct answer to your question?
This is the second time this week a gender chauvinist has tried to vaguely throw my own field at me as evidence for their ideological beliefs, without any credible sources to back up their statements. It's really starting to get on my nerves. Just saying "all anthropologists agree with me" is not a citation.
And accusing me of racial erasure, while accusing the mere existence of Mosuo society as a being some manner of fiction created by the "Leftists" of your own culture, is pretty fucking rich! Whatever else may be true, it is certain beyond any shadow of a doubt that the Mosuo consider themselves to be matriarchal, and consistently describe themselves that way. If you are accusing them of not knowing the "Facts" about their own culture, or of putting on some sort of a show just to appease Chinese tourists or something, you are the one erasing a minority culture's agency, not your imaginary opponents. The mere existence of "Brown People" (as you insultingly call everyone not like yourself apparently?) is not a communist plot.
