Agreed on the first part, that's where being a proxy for taking a walk comes into play. But regarding the second part: almost any lawn game would qualify for that, no? This could also be done over a game of horseshoes or Kubb. But that would miss all the other the gatekeeping, in-group signaling qualities mentioned.

Although maybe croquet qualifies as posh enough... Now you're having me wonder if we can categorize lawn games by the classes that play them?

https://en.wikipedia.org/wiki/Lawn_game