Why do vegetarians feed their animals food with meat

I'm confused why does a vegetarian not eat other animals when dogs and cats have some traces of animals in their processed foods, birds eat worms and sharks eat fish. Aren't our bodies intended to eat both meat and vegetables. Aren't we omnivores. I'm against animal cruelty, but I mean if your cat is eating traces of fish in it's cat food, your dog has traces of chicken in their food, and you have 100% tofu in your tofurky doesn't that seem kind of unusual. Animals are eating meat but you don't because you find it wrong. You find it wrong that these caged animals are being slaughtered left and right but then you feed them to your pets. I'm confused.