I'm always surprised to learn about such progressive countries which doesn't have separation of church and state. The fact that an organization have to prove it is a religion on order to issue marriages seems so medieval.
Where did you get the idea that marriages can only be officiated by religious clerics in NZ? That's an assumption you've made all by yourself.
In NZ, individuals can apply to be marriage celebrants, and organisations that "uphold or promotes religious beliefs, or philosophical or humanitarian convictions" can apply to be able to nominate their own celebrants. The article mentions examples like Yoga and Humanist organisations as non religious groups that have already done so.
And you can do the civil office thing too without a celebrant.