I'm sure the diaper didn't stay watertight - but after it filled up with water it was very close to the same density of the surrounding water (empty diapers don't weigh very much... full ones, dramatically do...) so it probably wouldn't have much affect on the flotation of the baby and may actually slightly help to keep the baby on its back. Statistically in adults I believe that drowned men float face down (diffiult to breathe) and drowned women float face up - this would be under "normal" ranges of body fat for sexually mature adults.
I'm not sure how an infant/toddler/pre-adolescent would float - I have a hunch it might normally be face down or 50/50. (Notice the position of the floating baby - some of the star pattern might be due to the water expanded diaper, but some of the small wiggles the kid is doing might aide him in remaining on his back (I can't stay afloat on my back without some additional motion)
What was impressive (so to speak... babies are much more trainable than most animals... and after a VERY short time they begin to try to shape their own environment and train any other nearby humans) was that the baby had been trained to flip over, right itself, and float. (I don't think the screaming needed to be trained - the baby/toddler in the video obviously has a few words and is probably pretty far along in terms of training his parents to understand those words).
Babies do still have more of the "diving reflex" than adults do - but from the video this is significantly more than that. From what I've read the diving reflex means changes in heart rate and stopping respiration while submerged. Nothing about positioning yourself so that the next breath is possible.
Even more fascinating was the footage of the infants who may not have been walking on dry land yet - the flipping face down, "crawling", then flipping over for a breath and a rest, then repeating was fascinating.
My only worry with this techique would be that it can easily lead to over confidence on both the parents' and the children's parts. Squirrels can swim fairly well when they have to, but my neighbors are still fishing drowned ones out of their pools every summer. Don't forget (Freakonomics citation) that swimming pools kill more children than guns by quite a large factor. Parents will still have to be vigilant, and pool access will still have to be controlled.
Still - babies are pretty amazing and trainable things... gotta go - someone is screaming for me.