For real - this saying came out at the advent of television. All the radio stars were flippin 'cause they knew they weren't going to be able to make the switch to TV, 'cause they has a "face for radio". Which is why they are largely forgotten.
Jack Benny was pretty good looking and made it to TV, but Fred Allen had a "face for radio", and never really made it. Did you know Fred was the Howard Stern of his day? He too was pulled from the air a few times.