This is outrageous behaviour if the story is as true by Air NZ. Definitely make a complaint;a and let us know how you get on.
At those small regional airports it is all usually very friendly given how few flights there and how small the planes are. Regularly rung ahead to say I am on the way and will make flight but will be a few mins after normal check in. Never an issue.
Heard that once at Taupo airport they held the flight for a few colleagues who took the same flight each week for about 40 weeks.... bad traffic on the way to the airport; last flight out on the Friday - didn't leave until they arrived!