Patriarchies are about 'ownership'; the ownership of goods, property, women, children, ideas, anything that is representative of, or produces money and power, and prestige. Matriarchies are about the preservation of society, the collective welfare of the community, and the easing of tensions. Patriarchies produce tensions within the society itself and between societies. The big 'I wants' take from the less powerful 'I needs' and this leads to increasing inequities. This inequity, in particular, can be seen by anyone who wishes to look deeply at Western, capitalist countries. The equity gap is widening, not lessening. The poor are poorer, the rich are richer and this situation is only getting worse.
We are on the brink of World War III because men who love patriarchy want to reinstate empire and women are suffering the complete erosion of their basic human rights and not just in Afghanistan. This world is not better off with men in charge. It's dying.
Matriarchy means a simpler way of life, a life that follows natural inclinations. It isn't natural for men to 'capture' women and make them 'belong' to them. Nor is it 'natural' for every child a woman has to be bound to the man who owns her. Children belong first and foremost to their mothers. This was the way of all societies until men learned that it was possible to limit a woman's choices in order to guarantee that any offspring carried his genes and only his genes into the future.
The Mosuo are not completely unique. They are not an aberration. There are other societies that follow the path of matriarchy. I think it is time we started to understand that it is possible to think and live in ways that don't embrace control and that don't use violence to enforce the will of a few over the many.