It all depends on the songwriting credit.
I think the logic is this:
Music performances on the radio are advertisements for the artist, and music performances that are live are generally done for pay by the artist. The songwriter gets paid for these performances, because he/she isn't there to get paid otherwise.
When recordings are sold, the artist gets paid, because people are buying THAT performance of the song.
It works out pretty fairly, in practice. The unfairness in most recording contracts is that royalties aren't paid until "label production costs" are paid. Until a recording sells in the 100k units range, the artist isn't going to make a dime. The way around that is to spend less producing a record, or hold out for a better contract.