That is not the case for all HOA’s. For some, they have gated communities.
Yes, but they still allow mail deliveries and visitors in some form or another.
After all, they aren’t building these amenities for everyone to use, just residents who either own property or are leasing property in a way that is approved by the HOA.
Yes, but there are other ways to manage that then setting up an HOA which can be expanded well beyond the management of that communal property.
You only have to look and see how other countries do it to see that HOA's are uniquely an American problem, one that has no justification in being as bad as it is.