I mean, roads are already open to everyone - your mailman can access them, visitors can access them, etc.
That is not the case for all HOA’s. For some, they have gated communities. For others, they are more than willing to enforce private property rights on people who don’t meet the “character” of the community.
If you extend it to ponds and parks and stuff, it wouldn’t be the end of the world for those to be public, either.
Sure it would. The wrong people might use it. 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.