Cats are obligate carnivores and natural predators, so it is in the nature of a cat to kill a bird or a mouse, and expecting otherwise simply confuses the cat. Cats haven't been domesticated for as many generations as dogs have. Sure, they have lived with people for a long time but until recent years, cats were valued for their skills as predators, and kept in order to reduce mouse populations. Even in my childhood, I grew up on a farm in Michigan, and I was the only one in the family who viewed the cats as pets. Oh, every finds kittens to be cute but no one else in the family would spend much time petting the cats, talking to them, or bringing them treats. They had a job to do.
Of course, cats are individuals and some of them are less likely to kill than others. My two older cats have never been hunters. I don't recall Cutie ever once even attempting to catch a bird or a rodent, and this included a couple of times when there was a mouse in the house. She would look at it and turn away, as if to say, "It's just a mouse. He won't eat much." Her sister, Lydia, caught a bird outdoors once, held her paws over it for a while and then released it to fly away, apparently believing in catch and release policies.
Ella, on the other hand, is a skilled predator. She has brought me mice, voles, and birds, usually quite dead. She doesn't even play with them. However, one day we had a mouse in the house, and she had been stalking it. Then, there was Ella at the bottom of the steps, while the mouse was on the first step. I couldn't understand how Ella could not know that it was there since it was very near her. Then she reached up and rubbed her head on the mouse, as if to indicate that this mouse was her friend. Later, she let the mouse walk away. But about a week later, there she was at the back door with a dead mouse. I hate it when she kills things but she is being a cat and it would be confusing for her to be reprimanding for doing what cats do.
So yes, if you are going to keep both predators and prey as pets, you need to make sure that are kept safe, and it is unreasonable to expect that predator to understand that the prey is family. Yeah, it does seem to happen, but I wouldn't depend on it. Many years ago, I had a cat and a parakeet. The parakeet was kept in a cage hanging from the ceiling, and not near any furniture that would allow the cat to have access to it. I had both the cat and the parakeet for a couple of years, and the cat didn't seem particularly interested in the bird. But one day, I came home from work to find that the cat had somehow managed to knock the cage onto the floor, and there were pieces of the parakeet strewn throughout the house.
