A lad resourceful enough to own and use an auger, would surely be able to make use of a rubber ring, or some such aid to buoyancy.

I still think reaction was his greatest problem.
Of course the law on action and reaction did not come into force until 1687.
Do we have a date for the song?
It could conceivably have come into force between him diving in and attempting to drill.
That would explain why he was unprepared.
If we could find a missing verse that mentions whether the ship recoiled when he dived....