Apologists for patriarchy and male privilege are among the most ridiculous and absurd defenders of any sort of unjust privilege that you can find. When you get right down to it, all their arguments ultimately reduce to waving around their genitalia and insisting that because their genitals hang down and outside their bodies, they have divinely-sanctioned authority to be leaders and deciders in the family, in politics, in business, and in all of society. So a penis is a badge of leadership.
The actual arguments they attempt to use lack any intellectual, philosophical, or moral credibility and this is because it's all just a smoke screen to distract attention from the fact that their position reduces to "because God gave me a penis." They don't notice this, though, because they are just so engrossed with their penis and/or are upset that others (mostly women, but also some men) refuse to acknowledge the leadership-granting qualities of the penis. Take one of their arguments and insert "Look, a penis!" and "God gave me a penis!" every so often to get a more accurate picture of what's really going on.
To be fair, not every defender of patriarchy and male privilege relies upon the importance of God giving some people a penis. Some defenses of patriarchy are secular and try to argue that male superiority is natural — as if the evolution of a penis was necessarily accompanied by the evolution of natural leadership skills. Secular patriarchy is no more rational than religious patriarchy, but it's a bit less honest because it refuses to acknowledge its own religious origins. It's like attempts to secularize homophobia as if the religious arguments could be secularized simply by replacing "God" with "nature."
I suppose that if we really thought that genitalia could be a sign of superiority, then wouldn't it be more rational to think that women were the superior ones? After all, their reproductive organs are inside their bodies where they are better protected. Shouldn't our leaders bit a little less vulnerable to a swift kick to the crotch? If God designed one sex to be superior, wouldn't it be the second one — the onecreated after all the flaws in the original model were revealed? "Whoops, those dangly bits are risky, let's try that again..."