Because if the one falling 20ft lands on a seesaw, the other side of it will toss two balls each of the same mass 10ft up.

Then 20ft should not be used in the explanation. They should just have one ball going at 2x speed hit the seesaw and have 4 of those balls go up at 1x speed.

That ends up begging the question, because the next step is "how high do you have to drop it from so that it's travelling twice as fast?" and you're immediately going round in circles.

Nothing of the sort. The seesaw can be in space far from gravitational influences. Potential energy is extraneous in this explanation.