I've been in Cutter's Cry a few times as a DPS recently because I am also leveling BLM and often get that dungeon.
I've never seen a healer do that before.
I think if the healer were to pull the adds away from the tank, the tank would get upset. I always thought the general idea for the tank to pick up adds and everyone DPS them down? That's how I've always seen it happen. Maybe different servers / data centers do it differently?
Also, because my group's DPS was just SO low there is no way they would have been able to kill all the adds before my Sprint ran out.