Scaling laws help.

I don't know how they work internally, but I can say that power hammers used for blacksmithing sound similar to the woodpecker in the video, only much slower. A couple hard, fast hits with absurd force before slowing down and stopping.

I'm guessing the woodpecker behaves that way because it's putting momentum into the hitting, even if that doesn't totally make sense in my head. When hammering on something, it's easiest to let the gravity do most of the work and focus your effort on aiming and raising the hammer, so you naturally have 1-2 hits that are solely momentum based at the end.

The woodpecker is horizontal, though, - not pecking in line with gravity - so my thought process isn't a perfect analogue. But if their tongue works like a spring then I can imagine it making sense.

Whatever energy goes back into the bounce necessarily isn't going into the tree.

The woodpecker is constantly applying it's force toward the tree so the bounce dampens over time.