In those days when radio and the recording industry were just getting started, you didn't have popular recording groups in the same way you did later.
Probably the equivalent would be a successful songwriter like Irving Berlin. I suppose 60s kids did have an inkling of who Irving Berlin was -- more from the movies and Christmas songs than from his earliest tunes -- but they probably didn't much care.
Of two alternatives -- everybody being fixated on the popular culture of 60 years back or young people never learning who Paul McCartney was -- the latter may actually be better. The problem is that they also never learn some things that are a lot more important.