In case any of you find yourselves in this situation, this is how it's done:
1. Dowload and print the application form from here: https://nz.usembassy.gov/u-s-citizen-services/how-to-renew-a-passport/
2. Go to the bank, get a bank check for the fee amount (in NZD) which for me was $165. There was also a $3 charge for issuing the check at my bank.
3. Find a post office that does passport photos. (My local office didn't but there was one a 5 min bus ride from me. Google will tell you what you need to know.) The downer here is it costs $20 for two little passport photos. To add to the sting slightly, you only actually need one of those photos. Save the other for posterity, I guess?
5. Staple the photo to its little spot on your application.
6. While you're at the post office, buy two prepaid courier envelopes (the smallest cost $3 each and are the perfect passport/application size.) Address one to yourself, and the other here:
Consulate General of the United States of America
Attention: American Citizen Services
Private Bag 92022,
7. Stick your application, bank check, expiring passport, and self-addressed courier envelope inside the Consulate-addressed envelope and mail away!
In total it cost me $194 NZD, and guys, a week later I got my brand new passport (and my old one with a hole punched through) back in the mail. I checked the tracking info against the processing date they had on the form they sent back to me, and they pretty much processed it in a day. I don't know if what I experienced was a unicorn, but it was pretty impressive. Sending kudos where kudos are due.