One thing I didn't mention was that when my daughter finally started swimming, she would swim underwater until she ran out of air and then stand up. The traditional school would never have allowed this, as you were supposed to breaststroke with your head above water at all times. Since she started to swim some distance underwater, it wasn't that much of a stretch for her to stay at the surface to swim longer distances.
The transition from dolphin to swimming underwater to swimming at the surface was very quick and seamless. Perhaps she was still helped by being drilled in the traditional school's proper technique, but I think we could have used the new school's methods from the start.