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.
It's interesting that people I know who went through a very formal swimming program as youths and then think of swimming as exercise and laps, tend to struggle with below-the-surface activities, like diving for items.
I remember my brother had a whole slew of underwater games we used to play like "tea time", "hold your breath at the bottom the longest", "headstand", "underwater flips", "drag your friend while they're underwater while you run in a race", "pull a gallon of dirt off the bottom", "upside-down Buddha" and so on including some pretty intense and dangerous-ish horseplay. There's probably a more comprehensive list somewhere as he didn't invent all of these but learned them elsewhere. Looking back I realize that we learned breath control, buoyancy control, how to handle strange orientations and direction underwater, diving, getting accustomed to handling stale air or out of air situations. But it all happened naturally through play.
Growing up, the only other kids I knew who could do the same kind of things also learned in similar informal ways built around play.
The summer when I was 5, my brother (an excellent competitive swimmer in his youth) brought me to a pool and we played a game called "jump and I'll try to catch you" and in the course of a couple hours of him moving to deeper and deeper water and farther away from the edge I basically figured out how to doggy paddle to him. By the end of that summer my brother had me fetching coins from the bottom of a 3 meter pool from a steady tread.
Only then did he start to teach me any formal strokes, but to be honest, because of how he taught me, I think of swimming as a perfectly natural activity like walking, where I decide where to go and at what depth and my body simply figures out how to get there. Specific strokes and techniques don't really play much part.
To this day, I love swimming.