This was being discussed on another thread recently - how some chickens will gorge on grain whenever it's available, but others prefer foraging and will wander off to free range while there's still grain in the feeders. I think the conclusion was that each bird is different, depending on the breed and the individual, and you can't really generalize.
I think it also probably has a lot to do with where they live, what kinds of plants and bugs are available and how much access they have to pasture, and possibly also the quality of the grain feed. But I think maybe omnivores are just generally more idiosyncratic in what they like to eat. When you eat everything, you can afford to have favorite foods that are different from your neighbor's.