Because dogs are expected to be kept on a lead. Cat's arent. I am referring to cats in this topic. But talking about dogs, in my experience from growing up with 3 hunting dogs (2 norweigan elkhounds and 1 nordic spitz) they have a natural instinct to chase. When dad let them off the lead it happened they caught a bird and killed it. Would he still let them off the lead? Absolutely. Everyone does it in Lapland, northern Sweden.
I will yell at OH if he is in, and he'll whack it hard over the head with either a brick or a chunk of wood. Sounds awful, but it's the quickest way to end the suffering. It is not nice, and I've cried when I've had to do it myself. But like I said, if you have a cat with outdoors access you have to either expect these things to happen, or keep the cat indoors! Having said that, some cats go through a phase of hunting and then get bored of it when they get older. Of course we hope this is the case with our Jameson.