Did they wave the cow over the head in the Bais Hamikdash?
I take my chickens home for the family, and give them food and water. They usually don’t touch it.
Sam, the Tur also quotes a Gaon, who used chickens.
Actually using money seems to be the latest addition. I wonder if anyone before the Chaye Adam mentions that. The point mentioned by Rashi is that you toss it. The animal, too, is killed as a Temura. Where does money come in to the picture?