My family talks about this with “Christmas” music. How many of the songs have nothing to do with Christmas but more with winter? Frosty the Snowman. Baby It’s Cold Outside. Jingle Bells. Etc.
The ‘drummer boy’ song is very popular, yet no such character is found in the gospel accounts (Matthew and Luke). The song was supposedly written in 1941.
Yet it is sung with emotion as though it is indeed part of the actual story, and is a key part of many Christmas plays.
If Winter Wonderland from Ziegfeld Follies of 1934 is a Christmas carol, then so should be When the Tumbleweeds Come Tumbling by Gene Autry (1938).