Unfortunately a lot of fraternal societies have disbanded and a lot of community organizing falls to women. Men and women community organize differently and traditionally men have had great luck in formal settings. For me, in our church (I know you're not religious), I lead our men's group (chapter of a larger organization), and there's a bunch of men who are the last people you'd expect to organize community but they're all doing it happily. But we're very hierarchical (even if jokingly so) and very formal (meetings, Roberts rules, elected positions, ceremonies). It's goofy, but I fundamentally believe men do well in this.

Unfortunately, I think it would be hard to form a men's group these days especially for secular people. But I encourage you anyway because otherwise it puts all the pressure on the women.

I have had several moms express gratitude to my guys for doing the work we do in the community because we also do things that the women would just never organize themselves. Which is fine... Everyone should play to their strengths.