[Andrew -- The proper spelling is "led", not "lead". A common mistake.]
Your response comes from the male mind To the male mind, all things are black and white, either (men run it) or (women run it). The upshot is that all male creations must have a winner and a loser. One man gets rich by taking from or bullying the less aggressive men around him. The man beats and abuses his mate (and children) for no good reason other than to establish his dominance (and because "they had it coming"). Men fight wars for no good reason except to beat down those who threaten them, and men see the threat in everything and everybody.
Except for those who only find identity in attachment to their mates, women (at least, those raising children) cannot relate to these "men's causes". Bottom line, they only care about keeping their children alive and fed and growing. They don't have to dominate other women for this to happen. The qualities women need to do this job are not competition and domination, but rather compassion and service and kindness . . . and the support of other women. The latter qualities are the ones missing "out in the world".
So should women "lead" men? Many of them already do, whether their mates recognizes that or not. (And I wish our patriarchy were not so embedded so that women could speak more.) But it is not a competition between men and women. Rather, we need a balance, which does not exist now, and will only come into being when men realize that their minimization of the feminine voice is the reason that everything men do on their own fails.