In the NZ context it's a wider part of the pest control discussion. NZ never had native land mammals (except a species of bat) until fewer than 1000 years ago, and everything's changed radically since colonisation from Europe began around 200+ years ago. We have lots of native flora and fauna that's in a downward spiral, being eaten or hunted or starved towards extinction. There's never been stability during that period, especially due to particular introduced species (rats, possums, mustelids) that destroy them.

Cats are also a big part of that dynamic, particularly feral, but it's a complicated discussion because so many people have grown up and still have them as pets. At the same time as there are efforts to reintroduce native flora and fauna to populated areas, the presence of cats is a contradiction, particularly when the law allows them to roam in ways that sometimes result in them being many kms from home.

The "I don't want cats on my property" line is often an extension of the belief that cat owners simply shouldn't be allowed to let them leave their own property in the first place. That isn't unprecedented, even near here. Across the Tasman in Australia there are lots of local jurisdictions which require cat owners to keep cats indoors or in proper enclosures. There are counter arguments, though, along the lines of "I keep my cat indoors at night" and "my cat never hunts any of that stuff".