I don't listen to music while shooting, but I wouldn't object to it, either, as long as range commands, etc., could still be heard. I mostly shoot on BLM land, not ranges. Like I said, I don't play music, but I could.
Music may take up a significant portion of your attention, as Restorer said, but I would think that's actually a point in favor of music-as-training aid. Not just external distraction, but the internal distraction. Operating a firearm and learning to either quell the internal monologue+external noise or have good, safe technique despite it, and maintaining accurate fire even with a bit of adrenaline amping you up from a song.
I imagine that if I had to use a gun in a life and death situation, there's a good chance that kind of a lot of stuff will be running through my mind: "What's going on?", "Where's he at?", "Can I shoot?", "Cover?", "Did I leave the oven on?", "I could really go for a scotch and a smoke", "Firefly was a great show." Who the hell knows.
My mind, and I suspect all minds, work in strange ways. In stressful (not really life or death) situations I've been in before, sometimes my mind wanders down some strange corridors. One that sticks in my mind was when an old man threatened me for trespassing on his land (I was 12 or so). Part of me was listening to him threaten to shoot me (gun wasn't with him). Part of me noticed that he was shaking, trembling kind of. Rage? Parkinson's? Just old? Did he live there alone? Did he have family? That's just the kind of odd turn my mind can take, even as another part of it was trying like hell to get me out of the situation.
I guess a lot of it depends on the person. If you're the kind of person who drowns out distractions with music, it can make you focus better. If you're the kind of person who focuses on the music, or for whom music+thoughts+physical activity clashes badly, maybe it's not such a great idea.