Meetup is kind of this. There were also a handful of these in the early-2010s wacky startup boom - I remember some acquaintances discussing Grouper, a startup that organized 6-person social outings.
I guess one distinction between those and your examples of country clubs, fraternities, and churches is that the latter are long-lasting institutions: you find a group of people, potentially spend an extended period of time interacting with them without commitment, and then you become a "member," which carries informal commitment. (In the case of cults there's formal/irrevocable commitment, but for the rest, you're free to leave, there's just a social norm that you don't do so casually.)
The flip side is that a lot of people have either first-hand bad experiences at such organizations or know people who have, and committing to an organization that turns out to be mentally draining (even if not abusive per se) is a worry. So maybe if you want to add some sort of structure on top, find a way to manage that risk. Certainly one step is to not accept as customers/members people who make your organization less fun to join, and be very up front about that. But beyond that, there might be ways to demonstrate to newcomers that the existing members are good people, or to support the organization in establishing social norms that discourage bad experiences, or possibly even figuring out what sorts of things toxic people tend to latch onto and designing your society in a way that lacks those.
(One other danger here is that the easy way to provide some of these benefits, like better career opportunities, is to be exclusionary and offer your members favorable treatment from each other and unfairly treating those outside the organization in a correspondingly unfavorable way. Country clubs, fraternities, and churches all have a reputation of helping the rich and powerful remain rich and powerful. It's possible to provide the benefits in a non-zero-sum way, by actually using the connections to make your members better people, but it's harder.)