No, a social system ruled by elites in a male-led dominance hierarchy is a patriarchy. You just said you are an engineer and not a sociologist (which I am) so please stop trying to tell me the definitions of words which you clearly do not entirely understand. A matriarchy would be a female led dominance hierarchy system — something that has never existed in the history of the world. Matrilineal/matrifocal cultures are based in balance between men and women and the importance of supporting (and not dominating) each other.
Primarily literally means "mostly, chiefly, or principally." You deciding that 75% doesn't meet the definition of that is your own bias in play.
A patriarchy is a Might Makes Right system where whoever has the most political or economic power typically calls the shots. Nobody said it's an absolute - in fact, I've been saying that part all along - that some men have more power than others because of the pyramid shape of the dominance-based hierarchy. In discussion of societal paradigms, pervasive dynamics matter because nothing is universal. 50 years ago rich, white, hetero Christian men had nearly all the power and privilege. Today, more women, more non-whites, more gay folks, and non-Christians have some positions of power but overwhelmingly, it’s still a social system of male-dominated elites at the top of the pyramid.
Perhaps it would help you to better understand how this social system arose in the first place.
“In contrast with the peoples of Old Europe as well as those of Mesopotamia, who worshipped a life-giving goddess that brought abundance, law, art, and beauty both the Kurgans and the ancient Hebrews worshipped a god of war and mountains, one who had no balancing female consort like that of the goddess. These invaders glorified in the death and destruction that they brought in the name of their god (Jehovah or Yahweh for the Hebrews), and in the case of the Kurgans, they actually paid devotion to their swords.
The one thing they all had in common was a dominator model of social organization: a social system in which male dominance, male violence, and a generally hierarchic and authoritarian social structure was the norm. Another commonality was that, in contrast to the societies that laid the foundations for Western civilization, the way they characteristically acquired material wealth was not by developing technologies of production, but through ever more effective technologies of destruction.
Eisler, Riane. The Chalice and the Blade (p. 86). HarperCollins. Kindle Edition.
This is what the term patriarchy actually refers to from a social science perspective — a male-dominated social hierarchy established and maintained through intimidation, coercion, violence, and fear. The war chieftains who are the most violent and most ruthless rise to power and prestige, and an ethos of Might Makes Right prevails. Culture becomes more authoritarian and much more hierarchical and stratified. Women and weaker men are at the mercy of the domination and violence of the stronger men. For the first time in human history, some people count and other people don’t.”
As the American Psychological Association notes in it's recent guidelines for working with men and boys, “Though men benefit from patriarchy, they are also impinged upon by patriarchy,” says Ronald F. Levant, EdD, a professor emeritus of psychology at the University of Akron and co-editor of the APA volume “The Psychology of Men and Masculinities.” Levant was APA president in 2005 when the guideline-drafting process began and was instrumental in securing funding and support to get the process started.
The main thrust of the subsequent research is that traditional masculinity—marked by stoicism, competitiveness, dominance and aggression—is, on the whole, harmful. Men socialized in this way are less likely to engage in healthy behaviors."
(Those are all dominance hierarchy informed gender norms)
Edit:
“Patriarchy is a social system, one that arose for the first time about 5–7k years ago. It is male-dominated, but that doesn’t mean that men always do well or flourish in it. That’s because patriarchy is not just a historically two-tiered genderized power differential, but also an entire system of social stratification and hierarchy that is established and maintained by intimidation, coercion, and sometimes even violence. At times there have been laws to maintain it as well. It’s essentially a Might Makes Right doctrine where the richest, strongest, and most powerful abuse others for their own benefit.
Thinking that patriarchy is about men and that you need to stand up for your “team” is really missing the boat. Yes, it’s a male-dominated social system where men hold most of the social, economic, and political power even today, but unless you are a guy who hangs out near the apex of the social pyramid, it’s really not about you, and in fact, you are just a cog in the wheel of the elites.
As I noted in my story about the historical origins of the patriarchal dominance hierarchy system, “Rather than defending patriarchy as somehow “my team” more men need to understand the realities of a social system that is steeped in blood, in authoritarian control, and in inequality — not just for women, but for anyone not able to keep themselves at the top of the hierarchy — either through accident of birth or through ruthlessness.”
