Yes that's exactly what happened. You can read about the taking up of songs by everyone when you read the diaries and news reports written at the time. Henry Mayhew's interviews and Emilie Cowell's diary are especially helpful this way. Mayhew's informant, by the way, lists the "Original Ethiopian Seneraders" at the St James Theatre -- however he says he might be wrong. His line up is the one on the picture (Germon's name as Germain -- but that's neither here nor there).
Funny thing is that we know Gil Pell's brother was there and he's not listed.
Back where I started. Fun discussion though.