It might be useful, in thinking what these hypothetical organizations would look like, to think about the history of the Elks/Masons/Odd Fellows/Knights of Columbus/etc. Which were essentially social clubs mixed with a primitive form of insurance; you joined the group and paid dues, and those dues paid for the families of members who became disabled or died, with the implicit promise that your family would get the same if it happened to you. Some of them were religiously-oriented, but most weren’t (explicitly), and they’re probably the most purely community qua community organizations that persisted.
It might be useful, in thinking what these hypothetical organizations would look like, to think about the history of the Elks/Masons/Odd Fellows/Knights of Columbus/etc. Which were essentially social clubs mixed with a primitive form of insurance; you joined the group and paid dues, and those dues paid for the families of members who became disabled or died, with the implicit promise that your family would get the same if it happened to you. Some of them were religiously-oriented, but most weren’t (explicitly), and they’re probably the most purely community qua community organizations that persisted.