> but most groups don't organically sprout around interests.
If you are only looking for people with the same niche interest, it will probably be more difficult to find local groups. But if you're open to meeting new people that are interested in social connectivity, local sports teams are the way to go. I've meet friends through co-ed sports that work in fields I would rather not come into much direct contact with: police officers, tsa, tax auditor, etc. People I never would have met otherwise (turns out they are just like other humans; who knew!) Seriously though, they don't all become best friends, but mostly social acquaintances and that works for me. I did get a couple best friends out of it some time ago.
> Sports, especially, are something that I have a very difficult time imagining enjoying.
Kickball. There are no high school/college/main pro leagues, so you avoid that one competitive team that all played on a school sports team. And get a bunch of people looking to socialize while playing a rather silly game no one is expected to excel at. To shore up that reality, join co-ed sports teams. I have witnessed games in both Mens and Womens leagues and they are much more about the competition.