I think you are talking about carnivores, not omnivores. Carnivores are like the big cats -- meat primarily.
Omnivores are like bears -- meat, berries, ants, whatever....
As far as human beings are concerned, In Genesis 9:3 God told Noah, "Everything that lives and moves will be food for you. Just as I gave you the green plants, I now give you everything."
You are talking about orders given to antediluvian humans. We are post-diluvian, and, as God told Peter, "Do not call anything impure that God has made clean."
We have omniverous digestive tracts.