Phil, I did some more googling and found this:
KC Courier News article on Stengel
The article describes his time playing for Aurora in 1911. It also has a gallery of photos, including one of Stengel with the KC Red Sox - while he's not wearing a sweater as he is in my photo, it's clearly the same player, and he appears to even be wearing the exact same uniform in both photos. The Baseball Hall of Fame could definitely be wrong, but I like their thinking.
Click on the small 'gallery' image of Stengel to the left to get a larger image and the following text below the photo: "Casey Stengel with the Kansas City Red Sox, one season before he came to Aurora. Photo courtesy~Baseball Hall of Fame"
This corroborates my theory which I explained to you by email. The postcard back is a 'KRUXO' with clubs in the corners, dating it to the correct years.
Also, I know I spotted a postcard of the 1910 Maysville team, but now I can't find it for anything.
Here is a young Brooklyn Stengel, the KC Courier image, and my image: