The obvious one is usually feral cats, which end up killing birds in large numbers if they haven't dealt with cats before. And a lot of those tend to be escaped house pets.
I'd argue that it's also a problem of biodiversity (at least in the U.S., don't know about New Zealand). If we hadn't eradicated almost all wolves, coyotes, foxes, birds of prey, weasels, and I don't know what else, nature would have solved the problem already.
Now that is hubris, what the hell do we know about long term net positives? Selfish warm feelings, while watching the reality of introduced extinction take place in a matter of mere decades. Pardon me.
Non-native introductions usually have the issue of such a sudden population explosion that the prey cannot adjust in a timely fashion and get killed very quickly, very suddenly.