The birds that still live in NZ are presumably capable of living in the presence of humans in disturbed habitats, if not in the presence of even more recent invasive species.
Sure, if anything NZ has a reasonable track record when it comes to eliminating introduced predators from select isolated areas. The conservation work that goes to protecting the kakapo is probably the most prominent example, but there isn't an unlimited supply of islands that can be used as breeding grounds for threatened species, so I understand the motivation here. New Zealand's native wildlife is some of the most beautiful, charismatic, and fascinating in the world. But let us never forget that the true culprits here are the humans who brought these invasive species with them. And frankly, I doubt John Key is in a particular hurry to dismantle the methane-emitting livestock industry - entirely non-native - and work to recreate the habitats that we destroyed for our own economic benefit. But one small step at a time, I suppose. Who knows what's next? New Zealand is always ahead of the curve when it comes to stuff like this.