I can't imagine asking someone to sing a cappella in an audition for a musical. You get absolutely no idea if they can sing on key, which to my mind is essential. You also have little idea about whether they have a decent sense of timing; there's a lot more liberty they can take when they don't have to interact with an accompanist.
You also have to watch the range of the singer, and where his/her voice lies. When I do specify a song, I specify it for each role in the show. On one occasion, we had an auditionee who learned about the show too late to learn the given song, so we just let her sing a song of her own choosing. She had a nice voice, and delivered it well, and was cast in the show. It was only when we well into rehearsals that we realized her two main arias lay right around her break; had we realized that at auditions, we likely would have cast someone else in the role.
Personally, I dislike cattle call auditions, though I agree that this seems to bring out the most people. It's harder for me to concentrate on what people can do, and if there are a lot of people I'm not familiar with it becomes more difficult for me to remember them after the auditions. I prefer to do small groups by appointment--about 6 to 10 people in an hour and a half slot. This gives sufficient time for dancing, singing, and acting segments of the audition, and also gives me sufficient time to pay attention to each auditionee.