That dead wood does not necessarily go to waste. When a tree dies it might not have green leaves but its life cycle is only half over. Dead and rotting wood is essential to the cycle of life in the forest. Standing dead snags get some bugs on them, then a woodpecker makes a cavity, then some other animal nests in it.... The peeling bark provides cover for those dreaded bats to sleep under.... Rotting on the ground it can be a source of bugs for foraging animals... It returns nutrients to the soil... Tops from a logging operation provide cover for critters like rabbits... Tops from logging provide some protection to new trees growing under them, allowing them a chance to grow in places where the moose/deer population is putting a lot of pressure on the forest undergrowth... (Granted down tops might constitute an excessive fire danger in certain places.)
And to stay on the original topic, I would still burn wood.