Imagine a small tribe of humans is getting into a war. They very easily could have men and women fight in the war, and it would help their efforts. It doesn't matter who's stronger. Now imagine that they lose most of their warriors in this war. If only men were fighting, then it's no big deal. They can go back to the village where women are waiting for them, and the tribe's population can be brought back up quickly. Now imagine if all citizens had gone to war. The tribe is reduced to a small number of each. And now there are limited females, who can each only have one, maybe two, kids at a time. The tribe would have a much harder time bringing its numbers back up.
Now, this is all ancient stuff, and the instinct may or may not be useful to humans today. Certainly, it is harmful in many ways; it is harmful to block women from proper schooling or work opportunity. There may however be cases where the bias can be useful.
People who insists complementarianism will argue that people should get their roles based on their sexuality because generally girls
cook better, knit better, and boys play soccer better, run faster.
But in my perspective of this topic, complementarianism just seems
like that thought is weakening the capabilty and confidence of each sexuality. People should be treated equally with their capability.
Complementariansim is just a thought that regulates meritocracy.
Egalitarianism is about rights, and that everyone should have the same ones. You can have an egalitarian culture, quite easily, that also understands that people are different, and have complementary strengths and weaknesses.
This is where the ideal would be an egalitarian meritocracy - equality of opportunity, with the best being chosen from those who wish to do something. In that way, you would understand that many of our roles are better suited for specific people, but that the specific person could be anyone with the right skills.
Perhaps you could call it a permissive egalitarian complementary system? The bottom line is that, if a person is good at what they do, they should be encouraged to do it. If they can't do it, a person who is better at it will win out in taking on the role.