I am from NZ, and have spend hundreds of days, with a rifle, wandering the mountains,
As Nube said, many of the large red stags are farm raised and released in to a restricted area or a controlled access area, i.e. big sheep station.
A nice even 12 or 14 pointer Red stag is consided a trophy by most Kiwi hunters,
I have seen as big as 18 point, in free range areas,
Some day, I hope to hang a red stag royal head, on the shed wall, ( 12 pointer so even, that a wine glass will balance in the top three tines.)
I don't like the multiple 30-40 point heads at all, think they are ugly, unbalanced and not a true repesantive of the animal, But, There is guys who do. sum very nice looking heads in the 25 point range,
NZ Hunting free range is very good, due to export meat, and wild game market drying up, currently the best hunting in 30 years, in many areas, but I already know of areas, with reduced trophy potential, due to excess animals.
Sika deer, Red deer, Fallow deer, chamois and thar, offer word class trophy, free range.
I really like Fallow trophy heads, I recommend you look into one, at the same time you do for a red stag.
The only outfitter I know is "wildside" in central north Island, there is tons down, south in the southern alaps, many are part of a working sheep station.
Wanaka, and South Canterbury areas, a great place to start looking.
Good luck, on thar and chamois, they are very tough trophy, to get on foot,
But sum unscrupulous outfitter, chase them with a chopper until too exhausted to run any further, before dropping of a shooter, to collect there trophy,
Distasteful, in the eyes, of pretty much all Kiwis, and moves are being made, to stop this practice.
If you get the chance to go out for a day fishing, I recommend chasing yellow tail king fish, epic charters out of Tairua, Corrmandle. have treated me and mates very well. I have twice come back after catching 20 plus 20-40lb kingfish, with arms so hurting, it was an effort, to pick up a beer. brilliant day out.
I see you had listed a couple outfitters, don't know either, but I have heard of Gary before.
The roar, (rut) for red stags is only a 4-6 weeks away, It may be a case of having to take, who ever has a spot, more than choosing.
You will be struggling, to do a good trip of NZ, with only three weeks, I think it may be wise, to pick one island, and cover it well,
South Island is much more scenic, grander due to the souther alps, North island for beach, dense, lush fern bush, and costal beauty. better sea fishing.
Chamois and thar, South Island only.
If you find your self in Taranaki, West cost North island. NZ, I am heading back for 6 weeks, Late March, and have a house there, (Bell Block) drop me a PM message, and I will show you around, Perhaps go for a quick shot for a goat, or a fish of the Islands,
Steering wheel belongs in the middle of the road, all other road rules are secondary to this one. Drive safe, kiwi drivers are aggressive, tailgating *****s.
southernman.