you're right. they are omnivorous.
i cooked some steaks on the grill the other night. i had a piece of fat about 4" long, probably 1/2" thick both ways that i had trimmed off before eating.
i took it outside where my rooster and 3 hens (and their pet banty, bridgette) were fartin around in the yard. i tossed it out to them, one hen grabbed it and took off, the others hot on her heals.
it was quite humorous, actually. they're light sussex, quite large, and definitely not used to that much exercise! they all started out running.... after 5 minutes, it was painful to watch her still trying to out run them. i say "out run," but they were all hardly even walking at this point, you could see the burn in their muscles lol