Don't over-complicate this.

1. DOT rules dating to 2011, require a refund to the original form of payment to be initiated within 7 days of a consumer's request.
2. DOT send NZ and all other US carriers and non-US carriers with flights to or from the US, a Warning Letter on April 3, making it clear that the rules apply.
3. NZ cares a great deal about DOT action. While it may not currently serve the US, US traffic is critical to its operations once Covid-19 passes. NZ, as a condition of its landing rights, agreed to abide by US law and is violating US law.

The process is quite effective. Indeed the consumer complaint is forwarded to NZ for a response. The key is that the only response is to make a refund as admitting that a refund has not been made would violate US law and subject NZ to substantial fines (up to ISD 24,000 per ticket).

Coupled with a chargeback, the process seems quite effective and there are multiple reports here on FT from people who have had fairly quick resolutions once they have completed both steps. For both, make sure that you submit a copy of the e-ticket receipt, the cancellation notice, your request for a refund and either NZ's denial of that request or a note that you have not been contacted. Make sure that you give NZ the full 7 days.

What is truly shameful is that NZ citizens must use a US regulatory agency in order to get what their own government will not do for them.