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, but it's not tied to any property ownership.
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 section of voters.
>> Want a playground or dog park? That's the government's responsibility? Maybe
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.
And the 7K person town where I live in, there are some conservation lands (no idea how maintenance splits up between town and conservation/commission and other volunteers), along with other conservation organizations. But dog parks, playgrounds, etc. just don't exist.
