It depends on exactly what you mean by independent.
I like https://rnz.co.nz/, but this is a government news source. It has editorial independence and doesn't have adverts so you know they aren't considering advertisors in their angles, but it is government owned, and has been threatened with having funding cut on occasion.
https://stuff.co.nz/ I believe is currently independently owned but it was owned by a big Aussie company until it wasn't doing well and the local CEO bought it for a token $1.
If you want a more Left leaning, non-mainstream source, then thespinoff.co.nz
There aren't many other sources I'd read, but others may chime in with others I've forgotten.
Its nice to think of us as a strong democracy but we have the same influences as the rest of the western world, so even if we have a nice voting system you could argue it's not a good democracy if everyone is manipulated by big tech and billionaires.
Edit: oh, I forgot to mention a lot of syndication happens, so it's common to see articles from one NZ news source on another news site.