In the Anthropocene around 90% of all macro-organisms will go extinct. Banning cats does not change this overall figure. Even for New Zealand cats are a lesser problem compared to the rats and possums brought there by humans and those are just a drop in a barrel compared to the overall effect of humanity. For example, New Zealands best solution to the rat and possum problem is bio-chemical warfare¹.
The whole "ban cats" movement is nothing but humanity shifting blame towards one of their oldest allies. New Zealands native birds are not endangered by cats, but because someone removed almost all of their forest homes during the last two centuries².
If you want to prevent the Kōkako and other native birds from going extinct, instead of protesting against other peoples pets, go and fund some breeding program or forest preserve.
At which point, why ban the humans?