Matrilineal Matriarchy.

There's a bunch of stuff on Native American tribes as well, you can find for yu'sen

Plains Indians weren't exactly matrilineal

With matrilineal descent and matrilocal residences, motherhood, not fatherhood, was central to Navajo culture. The earth, agricultural fields, and sacred mountains were all called mother, as were corn and sheep. Motherhood was thus defined in terms of reproduction and the ability to sustain life. Mothers were responsible for passing along Navajo religion and traditions to their young children and sustaining them through love and care. Three of their most central mythological characters were the maternal figures Changing Woman, Spider Woman, and Salt Woman.

Now there are two major asymmetries between our society and the one being envisioned here (I am going to ignore as merely cosmetic the notional liberation of women). And they concern childbirth of women, and the physical strength of men, that mean that matrilineal matriarchy is not a mere inversion of our patrilineal patriarchy.

A matrilineal society has no need to control sexual relations, because there is almost never any question as to who the mother of a child is. Patrilineal society needs to know the father of the child, and therefore needs to control the sexual relations of the woman. (Hence, for example, the 'value' of female virginity, still of vital importance to royalty and others.) The matriarch does not need to possess a man as husband, the way the patriarch needs to possess a wife. The pressure is off, wrt sex. A man's as well as a woman's loyalty will be largely separated from his/her sexuality, because their loyalty will be to the tribe of mother, sisters and nieces, while their sexuality will be external to this family. And a man's economic responsibility will be for his sisters' children rather than those of his personal begetting.

The asymmetry of physical strength means that in a matriarchal society, social control and status is not associated with physical strength to the same degree. It will be largely divorced from power and status, but remain probably a sexual attraction.

The notion of ‘societal control’ would be replaced by that of collaboration, rendering ‘loyalty’ less of an issue overall.

The matrilineal principle in Judaism does not mean Judaism is matriarchal, let alone feminist. Judaism is as patriarchal as all other monotheistic religions.

Basically, I presume matriarchy is both necessary and natural to the maintenance of a matrilineal society to at least an extent that makes it a noteworthy difference from Western culture which is patrilineal and patriarchal.

The notion of ‘power’ in matriarchal society would be more associated with qualitative potential or creative capacity than any quantifiable consolidations of potential or possession of value. The notion of ‘societal control’ would be replaced by that of collaboration, rendering ‘loyalty’ less of an issue overall.

The history of Iceland brings this into question. To this day girls inherit their mother’s name and boys inherit their father’s name. Women also used to fight, control ships and lead others.

You have not presented any kind of argument here.

In one instance the mother’s brother had more of a role in bringing up the child than the father - who was more of an uncle weirdly.

To keep on track with the topic how do roles differ between older men and women? Is there less difference?

It just may be that women that excel at being women suit positions that differ from men that excel at being men … the question is then more about what exactly we could possibly mean by saying ‘excel’ here and this inevitably cannot be disentangled from cultural factors above mere physiological processes.

The notion of ‘societal control’ would be replaced by that of collaboration, rendering ‘loyalty’ less of an issue overall.
— Possibility

Yes, except that collaboration is controlled. Not all control is coercive. I'm trying to imagine, without being totally utopian, how a a post-industrial matriarchy would function. Big institutions, national government would probably become less dominant, in favour of regional and tribal administration.

Mothers are loyal to their offspring. It kind of sounds wrong - too obvious to mention - I want to suggest, that the matriarch, in general, in a matriarchy, as opposed to the patriarchal matriarch one sees, is not an entirely separated self, identifying with an abstract body (oxymoron) as head, but as the birthed birthing of the extended family - I am the ancestors-and-inheritors ...* There is something radically different in the psychodrama of matrilineality.

*There is something of this in our (UK) current queen, even within the heart of patriarchy, dedicated to a lifetime of service as an almost religious duty. So old fashioned! So subtly different with the typical male identification of self with state that we call 'loyalty'.

Why is lineage an either/or deal?

However, it seems to still be about an either/or arrangement. Patriarchy v Matriarchy.
I don't think this is helpful. Indeed, doesn't it play into the fear of males that women are taking control and being less than subservient? Women have their roles to play; the main one being a mother?

such revolutionary offspring were all surnamed 'Wild'

Matriarchy is a stage on the way to patriarchy, to man's fullest realization of himself; he goes from worshiping Nature through women to conquering it. Though it's true that woman's lot worsened considerably under patriarchy, she never had it good; for despite all the nostalgia it is not hard to prove that matriarchy was never an answer to women's fundamental oppression. Basically it was no more than a different means of counting lineage and inheritance, one which, though it might have held more advantages for women than the later patriarchy, did not allow women into the society as equals. To be worshiped is not freedom. For worship still takes place in someone else's head, and that head belongs to Man.

To be worshiped is not freedom.

Fact: GARTHER, n. Men invented this elastic band to keep women from jumping out of their stockings and desolating the country.

man's fullest realization of himself; he goes from worshiping Nature through women to conquering it.

We could go into it a bit more if you are interested, but her views are not mine by any means.

So it is understandable that women felt that their very biological nature betrayed them, but I think they, and she, was wrong about that. So all the stuff about liberation from childbirth, I simply reject.

men escape the burden of pregnancy, and the risks and pains of childbirth.