They're running patriarchally-organized countries (and companies).

Are you using "matriarchy" to mean "women run hierarchical systems where decisions are made by authority and adherence is backed up ultimately by violence"? Of course women can use the tools of the patriarchy! It's made to exert huge amounts of power.

These systems, regardless of who ends up at the top, allow very effective organization to counter threats. They allow rapid mobilization of resources. They structurally reduce disputes over individual decisions in favor of having the struggle be on a slower timescale and be over who is where in the hierarchy.

This is true of hegemonic dominance hierarchies that run countries, and of those that run companies.

If you mean by "matriarchy" that "women run the patriarchy", that's a very different statement than "we should have a care-based largely non-hierarchical system (where the hierarchy rather than being an explicit reification of power is determined by social standing at the moment)". Or whatever you think a matriarchal society should look like that isn't a small tweak from what we already have.

Anyway, the point is that the "patriarchal" organizational structure scales and allows rapid, strong responses to threats. That's why it's less vulnerable to conquest by ISIS, Taliban, or any other small but aggressive minority. If a small minority opts out of nonviolent decision making, you're all set up to rapidly exert as much power as needed to get them back in line.

So what are we actually talking about here? I never quite know, because people mean such different things by "patriarchy"/"matriarchy". Elle is very passionate about a particular interpretation, which is the one I thought you were using since you linked so much of her stuff. (But even she very often seems to be using the gender-linked definition in her reasoning a lot of the time.)