Golf Clubs in The Villages--not the kind you swing.
 

Of the thousand or so clubs listed in TV there seem to be none about golf. Doesn't that seem strange?

Strange?
What would be the purpose of this club?

That, in a sense, answers my question.

Not the type of activity you need to organize a club - you need a text thread and someone to coordinate tee times.
There are ‘benefits’ of being a club (i.e. rent a room at a rec center) - not needed to golf.

Most neighboorhoods/villages, have golf text threads.

Not strange because you misidentified what you were looking for as it's called a golf group or league. The golf club of which you speak is a specific golfing venue that sometimes has a club house(but not all do as some small municipal courses only have a small shop to pay greens fees). Even in high school and college it's known as the golf team.

Now you do have chess clubs, archery clubs, etc. Softball groups are called "teams" and "leagues"(like bowling). Once you have this all figured out...notsocrates(or notplato)you'll have very little trouble finding what you want.

;-)