Joe: My guess is, broadsides were probably being sold at the time a song was quite popular, and was being sung in the music halls, and maybe even by the street vendors who sold the broadsides. So people would already know the tune, at least well enough to sing an approximate rendering of it.

When I was a kid, there were fan magazines aimed at teenagers, and sold from magazine racks in grocery stores, drug stores, etc., that printed lyrics of many of the then-current "top 40" songs in addition to articles about the performers. I don't remember the names of these magazines, but I think Tiger Beat is a more recent incarnation of the same genre. (At least I thought they published lyrics but I couldn't find any lyrics at their web site just now. It's been a long time since I actually looked at the magazine.)

Kids nowadays probably turn to the Internet for lyrics, but I'll bet the same demand still exists.

I figure broadsides must have been an earlier way of satisfying that same demand—not that the buyers were necessarily teenagers, but I suppose they might have been.