The ladies were to trek around breeding and bearing children in neighboring villages to stave off the dangers that come from inbreeding.
So the ladies saved the species. Men got to go hunting.
Not much has changed.
The study shows that the mitochondrial genome passed from mother to child was much more varied than that of the Y chromosome passed on by fathers. This confirms that in Neanderthal societies women left their families to go and live with other groups and bear children, while men stayed put in the same clan. It is a common practice in many current hunter-gatherer societies that prevents diseases and sterility associated with inbreeding.
My understanding is in many Native American cultures while the men went on hunting parties, "party" being the optimal word as they coi l d never carry enough food to feed the encampment or village, the women not only took care of the village, but fed the village hunting small game in the area. Men did and do overestimate their importance in society, l do believe.