Except that is not necessarily true.
If someone is butchering an animal, and tosses scraps to the chickens while they work, those scraps will be fresh (some people do this.)
And the list does not have any warnings about mice or rodents needing to be fresh. Someone could easily set traps, then take the dead mice out of the traps and give them to the chickens (so the mice would probably have been dead for some hours, or longer, at that point.)
If the issue is how long the creature has been dead, the list should say that.