Earlier quoted context omitted.
In the absence of an HOA, what you're describing as "shared facilities for a group of private individuals" would be commonly considered a private club. The difference is most private clubs that are not HOA's are not tied to owning specific property. For example, my sister belongs to a private pool club in their neighborhood. They pay membership dues to the club which provides operating revenue for the pool facilities…
Some things like beaches/pools/golf clubs/etc. can generally be policed with tokens/keys and so forth. That is not generally true of all shared facilities in a neighborhood. And I'm not at all sure the local government should be responsible for anything that residents should care to share on a communal basis. Want a playground or dog park? That's the government's responsibility? Maybe. But now that's up to a broader…
Yes, this is how it works pretty much everywhere else. Even my rural hometown with <1500 people has an elected park board that is responsible for parks, the swimming pool, tennis courts, summer rec programs, etc.