Eh, my cat has the disability of a bell around his neck. He's killed quite a few invasive lizards, but I can say with confidence he's never killed a bird (he's very proud of the aforementioned lizard kills, and would absolutely be showing off any birds he managed to catch). I've watched him fail hilariously at pouncing on a butterfly.
Bells are perfectly fine. I'd feel much worse cooping him up in a house all day every day.