Martha Raye's Song - Blues experts?

The movie was released in 1937. Yet Martha Raye has a song with lyrics about her mother singing Muddy Waters to her as a child.

So the composer and screenwriters must have known Muddy Waters as a swing reference. Forget about the cradle lyric, taking the story back to 1917 or so. Instead, just assuming everyone would know Muddy Waters in 1937, I don't get it.

Evertyhing I've ever heard, or researched, about Muddy Waters indicates he was not known outside his home state until the 1940's - 1941 was when he first traveled and made a record. What is his real history?