From my research, there are a lot of options in NZ. My long term plan....
The best is to be flown by chopper to remote mountain area, stay in public mountain lodge, have one week hunting walking and stalking, climbing.
You and (eventual) guide, have to take care about your food (either canned, or something you shoot), after a week, or as per agreement, chopper collects you back.
There is public database of public mountain lodges and they can be booked online
Option no. 2 rent a 4wd, go to the mountains, find a public lodge, and do the same.
Option no 3, do the same under tent.
Then remaining options are farm based outfitter, who can offer estate hunt, or hunting trips in public land, by chopper or by 4wd.
Shooting tahr and deer in New Zealand on public land is free (trophy fee - zero), but there will be outfitters charges.
Shooting tahr and deer on fenced property, you pay for each head.
Shooting from chopper is possibility. But only one of the possibilities, not the only one.
If you want to organized DIY hunt on publc land is possible, but it takes a lot of research and prep. basically models 1 to 3 above
The best advertising and arrangements you can find on internet are fenced hunts (most expensive), everything else needs more deeper research. Large NZ stags, with massive antlers are usually in fence.
I havent done NZ hunt yet, but close friend of mine who lived in Australia did this frequently