we would see that graphic used on a Country Club of Virginia item dated to a similar period of time.

I typed in "Country Club of Virginia, Richmond" into Worthpoint.com, which has a free archive of all Ebay auctions from the last 15 years. (You need to subscribe to see the auction results, but seeing the auctions and the auction images is free.)

https://www.worthpoint.com/inventory/search?offset=0&max=20&query=country+club+richmond+virginia&categories=

I added Richmond to the search because there is also a Country Club of Virginia in Long Beach. Go figure.

It appears that in the past the Country Club of Virginia may have used different logos, than the logo on their website. Note the match book and the pennant at that link.

It is possible that the logo on the website is something created by a graphic designer much more recently.

Maybe one of us will find a vintage item with a similar logo.