People like to make the case that humans are omnivores. Now literally, that means "eats everything", but in the textbook sense, it just means we are not strict carnivores or herbivores. We are adapted to eat some vegetation and other animals, not "everything".
To prove this to yourself, imagine being dropped in the wilderness, anywhere on earth away from civilization. I live in Montreal, so I like to take the example of a boreal forest, typical of my latitude.
In the summer, the land around you would be teeming with life, but the greatest portion of it would be unavailable to you: bushes, trees, and grasses -- you couldn't digest maple leaves or pine cones. If you were lucky you might find a few wild berries. If you knew what to look for, you might find some edible roots and herbs. If you were quick enough or smart enough, you might be able to catch a fish or bird, or other wildlife -- in other words, you would face the same challenges as any hunter-gatherer. And you would be eating the very foods your body was designed to eat.
So remember that the omni in omnivore stands for: a preciously small fraction of what we can digest and thrive on. Moderation need not apply to this group of foods. In fact, if restricted to these food choices you could eat as much as you want. This is what our ancestors did, and they did not suffer from obesity and other "diseases of civilization".
When we talk of moderation it's usually in the context of modern (post-agricultural) food, or just things which can make you ill in the short term, like alcohol, or sugary drinks. But then you run into the problem of figuring out what a moderate amount is. Is it two beers a day, or three? One Coke or two? What is a moderate amount of rice or other grain? Of saturated fat? Do we judge by how much we can tolerate in the short term without becoming ill, or is there a long-term risk?
I believe that when an animal eats its natural diet, its portions are moderated and regulated by its body, and the feedback mechanisms which govern hunger and satiety. This is what we observe in nature.
Feed it something it is not adapted to, in sufficient quantity, and it will ultimately get sick. This is what we observe in modern human civilizations.