You hear the "birds don't have plastic feathers" line because the fletchings on an arrow are there to help it "fly" straight to the target. That's the only reason feathers exist... for flight.

A bird wouldn't fly too well with plastic feathers.


As far as the other things go, none of them occur naturally in nature like feathers do. And none of them transfer directly from the natural state to the "arrow" state like a feather does.