Because radio stations' DJs would say "and here's artist with song name", I firmly believed that one of two things must be true:

Radio stations had the bands always playing live for them

And/or

Radio stations would present a local DJ, but they would then tune in to the artist playing the track live for a larger station

What's even funnier about this is that my dad has at the time taught me how to use a tape deck to record radio and CDs to tape. So I clearly understood recording mediums. Just, the idea still lingered in my mind for a while.