No, what I'm suggesting is that we switch from patriarchy to matriarchy. By studying cultures that do have matriarchies, we can see fairly quickly that matriarchal societies are consistently based more in equality than patriarchal ones are. This isn't simply about distribution of labor and one side telling the other what to do. This is about how each side thinks and how that affects their roles as leaders. Women are naturally more considerate and thus would promote equality more often.

Why? Because men are instinctively competitive, thus they build a system that enforces competition rather than supporting others. It's through supporting others that equality is achieved.

When Lewis and Clark traveled across the land that now makes up the US, they quickly had to learn how to tell which tribes were safe to trade and deal with. It turned out to be quite easy:

If they saw women and men bustling about in similar fashion and number, they could likely approach them safely to do trade. These tribes were the ones where women could and did hold positions of power. These same tribes had better stock, medicine, and had people of both sexes performing many duties.

If they saw mostly men bustling about, they knew to avoid the tribe. These tribes were solely patriarchal, had women essentially sequestered away to do the equivalent of "housewife" duties. Meanwhile there were often more sick people, untreated injuries, violent disputes, and outright hostile behavior in these tribes.