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.
Sawdust is used on some floors to absorb any liquids that fall rather than trying to prevent them being spilt. The sawdust is swept up and replaced each day. This was common in the past in pubs and is still used in some butchers and fishmongers.
Sawdust has a variety of practical uses, including ... as an alternative to clay cat litter ...
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.