While I understand your point and need, I want to give you the other point of view.

You have explained why men need women. But many times the same men who needs women, is not treating them well. Or is a perfect extranger. As is normal, woman don't want to spend time with someone treating them badly. In fact, how many times a woman who is abused have been told "why didn't you leave him?"

Other times, you are at the train or the underground and a perfect extranger unfold his lifes and problems to you. And the thing is, why is this expectation that you offer emotional support to someone you don't even know? Best case scenario is weird. And what kind of help did they expect from a stranger in a 30-60 minutes commute? Let me tell you that it is quite uncomfortable having a stranger telling you the ugly things in his divorce while saying that all women are whore while being a woman. This has happened to me and is messed up.

This is the thing, instead on focusing on conforming to absurd expectation about keeping their feelings to themselves and treat women as crazy when we suggest that men should be able to talk to both female and male friends, you might want to take that advice. Yes, no one likes to feel vulnerable. But if you can't trust your friends to listen an support you, they aren't your friends. And to hell with anyone who call you names for not conforming to stupid gender norms.

You won't be less of a man for asking for help and support from family and friends (regardless of sex).

And another tip, the woman who are willing to emotionally support men, expect the same in return. It is not right that you ask for women to be understanding, but that you don't do the same in return. Even if in your head you are offering something else. Because that something else might not be something that woman wants. For example: offering sexual attention in exchange for emotional support might even be considered ofensive if the woman only see you as a friend. Connection requires effort on both parts.

Maybe is about what you value more, emotional support or the image other have of you. Sometimes you need to have the courage to say "I don't care what other people think, I would reach out for help". In fact I bet that you do know when a (male) friend is having a bad time. You don't talk to him to not embarrass him, but you know. Well, maybe it is better that he initially feels embarrased but loved and supported than alone and with pride. Besides, eventually you would develop the skill to offer support in ways that won't feel bad for your friends.

Everyone's life is hard, women's life too. So start by supporting the people you care about. They would take care of you in return. And it would make you feel better.