Deme

Impulse is the integral of force over time, but I get what you’re after.

When pecking at a tree, the maximum force exerted is what pushes the wood beyond its breaking point. That maximum force can be increased by increasing the impact energy as a whole (wasteful and costly) or shortening the impulse. A woodpecker isn’t trying to do soft blows to shake some branches, it’s trying to shatter a small portion of the trunk, much like someone looking to shatter their opponents nose would choose bare fists over boxing gloves.

IIRC this theory was debunked some time ago by a study. If you think about it, any dampening within the skull would lessen the force of the pecking and the bird would have to hammer away harder.

Edit: Link to a study