There’s much less drag if you’re completely submerged, allowing you to go faster. But its hard to use your arms effectively without being able to recover them out of the water. So, over short distances, dolphin-kicking underwater will be faster than all the other strokes. Not because your legs are more efficient in general, but because being underwater is more efficient and it’s hard to use your arms underwater.
The problem, of course, is breathing. The restrictions on how far you can go underwater are there to prevent competitive swimming from turning into a breath-holding contest.
Nitpick: back when butterfly was still considered a variant of breaststroke, the butterfly arm motion was allowed, but the dolphin kick was not allowed. (Competitors at the time used the butterfly arm motion with the breaststroke whip kick.) It was only after butterfly became its own stroke that the more efficient and faster dolphin kick was allowed in competition.
Exactly. The same thing led to a rule change requiring breaststroke swimmers to bring their head out of the water for each stroke. In the 1956 Olympics, breaststrokers were swimming the whole length of the pool underwater. They used their arms and legs, of course.