The reason it works is because most men have innate urge to protect and provide for women.
This is the irony behind it.
Another irony is that patriarchal societies are brutal on men. Men are expected to bottle up all emotions and illness, never take a sick day off and end up with mental breakdowns or bei g abusive (and this victimises women and children of course).
So, what is actually optimal is something I can dub as soft patriargchy andnis what some ofnthe west is ending up with.
Soft patriarchy happens because even when equal opportunity existe far less women want high stress or high physicality work.
Notice that 99% of construction workers are men and 0 women are complaining.
Notice that when STEM other then nursing ends up with heavy male sway, women do complain and even men do, trying their best to force 50/50 by a variety of ways, some forceful (affirmative action) others subtle (constantly pressuring girls to grt into STEsm if they want to feel fulfilled as women ans feel they did all they could for their gender).
The reality is even when we talk pure hierarchy, the stresss it takes to endnup as an executive of a company truly attracts much more men that it does women.
Not only that but men barely ever seek out high career achievement in a spouse whereas many women want either a high flying career man or at least a poorer man that is dedicated and has some form of stable income and job.
This means when men and women full do fight against patriarchy, you will find that women themselves suffer with a system that makes men less appealing to them on average and pushes them to experience stress levels only men are designed to endure.