Not at all. Where I live if someone's animal is on your property you're absolutely allowed to shoot it. If the owner places so little value on their pets life that they let it run around unsupervised, who am I to place more value on it than they do?
Small mammals is kind of the point. There are mice in the garden that try to get into the house; the cat's job is to kill as many as possible.
Regarding birds, I'm not sure. It can't catch magpies and other crows. We don't see or hear other birds (we live outside Paris and don't have that many birds). I don't think it would make a lot of difference to keep our cat indoors since all other cats in the neighborhood also live outside -- except upsetting and depressing ours.
A few weeks ago I heard the claim that you have to supervise your cat while it is outdoors or else you will be responsible for extinction of certain species... That sounded a bit ridiculous to me so I read a few articles on that topic. (the first few sources of https://en.wikipedia.org/wiki/Cat_predation_on_wildlife are a good start)
While cats do kill a lot animals, including endangered species, many articles that claim your outdoor cat is irreversible destroying wildlife seem quite sensationalist. Many articles make it sound like pets are the real threat but only mention on the side that feral cats are responsible for the majority of killed animals instead. It would be far more beneficial (for wildlife, pets and feral cats) if people would support projects for controlling feral cat population than confining their pets.
We brought a cat inside that was free roaming and an amazing hunter. She was so effective and cruel we couldn't take it. Half eaten lizards, wrens on the wing, a wounded hummingbird. That was it. She's now a fat sleepy lap cat. God bless her.
Of all the low hanging fruit on the "stop destroying the planet for no reason", keeping your cats inside or on a leash requires basically 0 effort.
https://www.smithsonianmag.com/science-nature/moral-cost-of-...