I've never seen sh'chita of birds, but my understanding is that kisuy hadam is done with sawdust. Why sawdust specifically?
I don't know if this is the only reason, but presumably one factor is the fact that it's good for clumping up liquid messes for easy clean-up.
If I were operating a slaughterhouse, I'd rather deal with sweeping up clumps on a daily basis than frequent mop-ups of bloody, muddy messes.
Any material that seeds will sprout in is acceptable, including, ashes, sawdust, gold dust, etc.