Well if you have an outside cat it's kinda hard to keep an eye on them climbing trees and killing a songbird. If you want to climb up the tree after the cat, and stop it, be my guest.
Edit: And then there's cats such as farm cats, too, which are /supposed/ to kill things like rodents.
Train them...like really. If you want a pet it should be a more arduous process to get one imo. I live in a place with a very dense amount of endemic species. Irresponsible owners do not deserve an excuse in this location. The damage their pets do is ridiculous. Don't want to be responsible? Do not take on more responsibility. This also applies elsewhere. Pet owners can be very irresponsible. Guess it is okay, our entire species is mostly irresponsible when it comes to the environment.
About the Edit? Sure, but keep them on the farm. Honestly it isn't those types of pets that are the biggest nuisance. They usually stay on the farm, get trained, and don't go sneaking into areas they shouldn't.