I agree, but that part wouldn't apply because birds aren't pets (unless someone let out their bird, which I don't think happens, but in that case yeah it should apply).
This should be an offence, I hate seeing people with dogs that they can't even control on a leash, it's a danger to everyone and maybe this would make people think again about getting an upgraded pit bull terrier with max enchantments.
Also I would never let a cat outside without full supervision, they have become an invasive species basically (as u mention) killing off lots of other animals whenever they are out (but this should still be an offence if it was killed by someones dog, etc and the other way round).