Why wasn't Bilbo famous, by the time of LOTR?

(Note: I'm trying to avoid spoilers, as much as possible.)

Toward the end of LOTR, we are told that bards have already begun singing the tale of "Frodo of the _____ ______." It took the bards roughly a week or two.

So, why wasn't Bilbo the Hobbit famous, by the start of LOTR? I would have thought that the tale of the "The Hobbit" (considering the monumental end results) would have been part of every bard's repertoire by that time, putting Hobbiton squarely on the map (so to speak). But instead, Sauron's black riders had to go searching for the hobbits.

Answers based on canon obviously preferred, but I'm OK with educated guesses.