That's not wrong, that's not true either, mostly that's pseudo-scientific naturalism. For exemple a naturalist would say that the Lion is the king of the jungle, therefor according to ideological anthropomorphic projections from humains, the male Lion hunts for food and is a predator.
Nature not being "stupid" or doing pseudo-science, it turns it's completely false: female lions having a higher ratio of muscle to fat therefor being more agile, and having no mane therefor being more discreet, are the ones who mostly hunt, while the male lion having a lower muscle to fat ratio but being heavier, is the one better equipped to "stay home" to protect the kids and the territory and even evolved a big mane to intimidate away predators.
So your "survival" bullshit has nothing to do with science, anthropology or even zoology, it's just simpleton ideological projection. However current women's and men's brain are indeed wired differently on average, and there are reasons that don't have much to do with genetics which are just the encoded pass-through of conditioning, experience, health etc...