It's interesting that those Edison recordings on the UCSB site include
Hail Smiling Morn and The Mistletoe Bough, the same two non-carols* that are still attached to the "Sheffield" traditions.

* Hail Smiling Morn has nothing whatsoever to do with either Christmas or Christianity. The story of The Mistletoe Bough starts at Christmas time but that is the extent of the connection.