To me, this is a little like not allowing golf carts, but, on the other hand, I do understand it. Many visitors to a country club aren't there for the golf. They're there for a lunch or even perhaps a wedding.
While I couldn't care less, I'm guessing that there are some of those folks that would find shoe-changing to be a bit clumsy.
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Doug - wear sneakers at my place, and you may find yourself doing a butt-plant when your fleet slip out from under you walking down a fairly steep hill in the rough. Then again, a kid from North Carolina shot a 67 during an AGJA event here a couple of years ago while wearing sneakers.