Pets aren't people, they belong in the baggage car.
No, but they are living, breathing creatures, so I would not want them somewhere that isn't climate-controlled and somewhere I wouldn't have access to them to check on them to make sure they are okay. If you wouldn't put a human in a baggage car, then an animal doesn't belong there either.
Do you get to check on a pet on a plane?
I have never taken my pets on a plane. If I did, they would be under my seat.
If they were not allowed under the seat and had to fly in the cargo hold, I would not take them on a plane. You know why? Because they are live creatures and should not be treated like suitcases. Like humans, they are sensitive to cold and heat and need to be fed and watered.
Additionally, if there were an emergency, I would have no way to rescue them (from the cargo hold or baggage car). If they're under my seat, I can grab them and go.
My original point remains: humans do not travel in cargo/baggage, so neither should pets.
When you have a pet, you are responsible for its health and safety. Anyone who has the attitude "it's just a dog/cat" should never, ever own a pet. Pets are not a piece of furniture or a decorative item. If you wouldn't put your kid in the baggage car along with some food/water bowls, why would you put your pet back there?