It's not just about the severity of COVID when it comes to pets. Like a previous commenter stated, COVID can jump ship, infecting animals as well as humans. While infected, animals can help COVID mutate, and due to the fact that it can then jump back to humans, we risk contracting a completely new COVID variant at worst.

Edit: A reply right below this one said that due to a mutation in their ACE2, dogs are resistant to COVID, as that mutation doesn't allow COVID to multiply efficiently, thus reducing transmissibility between humans and other dogs. As a result, my reply doesn't really apply to dogs.