My house, though, is not common property. Let the HOA rules cover only the common stuff.
Incidentally, local government where I live provides tennis courts, basketball courts, swimming pools and plenty of playgrounds. A few of these are within walking distance...
If it had a convenience store nearby, it would be like a little village
I get that there's a certain type of person who'd like this, but for the majority of people, you're highlighting the reason people don't like HOAs. You're creating a little city, whereas most people just want a home.
In any case, as I may have said somewhere - if you exclude condos, only a small percentage of HOAs have any shared facilities. No courts, no clubhouse, no pool, etc. When I was house hunting, easily under 10% of HOAs had any shared area.