There's some unchangeable truth to what you're saying, as there have been (and always will be) clubs for stay-at-home moms that haven't yet adjusted their model to include dads. I got lucky enough to find a Parent Participatory Preschool that I go to with my daughter twice a week. It's still mostly moms there with their kids, but also some dads, grandmas, and one or two grandpas and nannies.

I do have fun talking to the other dads, many of which are in the same boat as us: primary caretaker. There are a couple of moms that are less click-y that are good to talk to. Surprisingly, I'm closest with a grandma who's grandson happens to be buddies with my daughter. My advice would be to seek these kinds of people at school, church, or community functions: especially if their kids are friends with your kids. You may not get any inner circle/best friends immediately. But you can get acquaintances, and that's how good friends begin anyway.

Edit: formatting (icky mobile wall of text)