> Why won't the men compete for the jobs the women are getting?
Because different people are suited to different jobs. Why didn't (and don't) women generally compete for manufacturing jobs? Because being a creative designer or HR professional or pre-school teacher or dental hygienist is something much preferred for certain types of people. Just look at the numbers [1].
Men not only don't want to do certain types of jobs, but in many cases can't - they're not well-suited for them. So even if/when they compete, they'll lose to less experienced women just entering the workforce. And vise-versa for other types of jobs with flipped genders. There are gender differences, plain and simple. You can argue why they exist, but for this discussion it's not as relevant at this point.
I would like to note that the next "industrial wave" could affect women pretty hard as well: artificial intelligence. Recruiting and HR software, health-care robotics technologies, therapist chatbots, etc, could upend the female-dominated jobs listed in [1].
> Alternatively, the men could stay home and take care of the kids while the women work at the jobs they have, providing the stable two-parent household.
Heh, I guess you haven't read many women's online dating profiles for what is acceptable in a mate and what isn't. You generally have to be higher up on the ladder (education, wealth, status, etc) to attract a woman for mating in the first place. How this manifests in data: 80% of guys are "below average"! [2]
But who needs data, ask your grandfather if he could have "stayed home and taken care of the kids while the woman works" and he'd laugh at you and tell you to stop being a sissy. They may not have been able to explain it, but older generations have typically understood more about human nature than we give them credit for.
[1] https://www.dol.gov/wb/stats/occ_gender_share_em_1020_txt.ht...
[2] https://theblog.okcupid.com/your-looks-and-your-inbox-8715c0...