"Patriarchy" hasn't been designed...full stop. Cultures never are designed, they evolve as .groups of people employ technology and common behaviors to solve the economic challenges they face.
Currently, our culture is adapting to the new economic reality that sex isn't a huge advantage in economic success any more. This adaptation will take time of course, every action spurring a reaction, but it seems irreversible.
You ask if men hang out with women when there's no chance of sex. The answer is yes. My close friend is skilled at river life, from reading the water and rowing to making and breaking camp. These are fairly rare skills, and given how much tougher and less fun a solo river trip is, i value them highly no matter who possesses them.
She is also a talented musician, so we often play and sing together. Once again, these skills are not sex related, and sexuality plays no role in this facet of our friendship.
You ask about female role models...my mom is the key role model in my life, and not because she was sweet or self sacrificing (she wasn't), but because she insisted in following her own heart, no matter what the cost.
In her case, the cost was a marriage, some dearly close family members, and a certain amount of social disapproval. Authenticity simply was more important to her, as was creativity and self actualization. In each of these areas I have tried to live up to her example.
My dad, on the other hand actually WAS a hero (WWII), and I did my best NOT to follow in his inauthentic footsteps.
Your piece is stuffed full of stereotypes, and while stereotypes gain recognition through repetition, they do NOT determine the paths that individuals choose.
If you don't know women who inspire people of all sexes, don't draw in friends and life students of all sexes, then that is very sad.