I suspect Brygun HAS cut branches without picking them up immediately: The pick branches quest has you pick a lot of branches to be delivered, and I don't pick them up until I've cut all of them.
I've also used branches as building material cut sufficiently close to the building site to be used directly from the ground (shelters and pit traps), like JP_Finn does.
I'd be careful about pushing things around if the stacks are large, though: I made the mistake once of pushing the feather collection to one side. That took many hours, as pushing consumes time per item, while picking up stacks takes time per stack. Pushing stacks of many items onto animals can also take a silly amount of time.