Well i always thought Eb, Bb and F and the relative minors were popular because of brass instruments, and Clarinet LoL. Unless you own a C melody sax.
That could certainly be the case. My knowledge doesn't extend to these instruments. But it was the era of the big band so it makes sense.
If a piano plays in the key of Eb, though, Bb instruments would transpose to F and Eb instruments would transpose to C. It seems that Eb concert pitch could be useful here, too.