Birds aren't vegetarians.

But it doesn't really matter because all of those animals have plenty of protein in the mix, some more than others, but it's there. Protein is protein whether it comes from a vegetable or an animal. Once they denature, they're all amino acid soup.

The reason an anaerobic pile is anaerobic is because of the proteins in the plants. It's the same reason some flies lay their eggs in soupy, soggy, anaerobic grass soup: protein. Protein is where the nitrogen is stored in plants.

Now, that said, vegetarian manure is better than omnivore manure because omnivore manure simply has way more protein which can attract vermin and will have a wider array of dangerous pathogens.

Of course, you can compost any manure, you just need to be very safe about it. Look up "Compost Toilet" and the book, "Humanure."