Because it still puts men on top in most ways, even while it hurts them too.

And it definitely applies to all western societies.

You can see it in this very story. "Men are strong, they don't need help. Women are weak and emotional, that's why they need support." Yeah, it's devastating for men in this situation, but it's the same logic which makes people say men are natural leaders or whatever.

Because they didn't get over the societally enforced misandry and its dehumanization of men.

The Patriarchs in question does not refer to all men. It specifically calls out the culture of the "elders"and those who have had ample opportunity to become established influences through the system. Those who subscribe to old fashioned beliefs and those at the top of the power structures that benefit from their compliance from younger, poorer generations. Emotionally distant men and limiting variability within the group makes the entire demographic more easily exploitable. A lot of roles in the family and society exist for men outside the title of "patriarch" but patriarchs specifically use their role to self legitimize their power over other people and make everyone in some way subservient.

It's kind of a shorthand for "old fashioned" conservative systems of organization that prime men to be "leaders in embryo". The gendered component is still valid because it is still a dominant model that is marbled with minor subversions of it. Women and non-standard men may have changed their place in the family but even when they reach the top they have to make themselves non-threatening to the cohort of established powers and play by their rules to succeed.