First, recall that impulse equals the change in momentum.
FΔt = Δp
With or without the cushion, the change in momentum of the pile driver was the same. It’s momentum just before impact was p, after the impact was 0, cushion or not.
With the cushion, however, the impact duration is lengthened (Time is bought by allowing the egg to back away from the pile driver) For the same change in momentum, the maximum force is reduced.
See the F–t graphs below. Keep in mind the area-under-the-graphs for both graphs should be exactly the same, since the impulses for both cases were the same. A longer impact duration allows the impact force to be smaller and yet bring about the same change in momentum.