"Neither of the Contracting Parties shall be bound to deliver up its own citizens under this Treaty, but the executive
authority of each shall have the power to deliver them up, if, in its discretion, it be deemed proper to do so."
So there's a bilateral opt-out for both parties, at least in the case where their own citizens are involved.
The problem with that is the asymmetry between the US and New Zealand in terms of power and trade. Having both parts being able to opt-out effectively means that the smaller part will be free to do exactly as they're told, or else.
I hope New Zealand deals with this case and does so fairly and ignoring external pressure. Sadly, this looks unlikely.
New Zealand has a history of standing up to the US, especially its military. See the fallout from declaring NZ a nuclear free zone which forced the dissolution of the ANZUS military treaty between the US and NZ.
So, are you arguing that corporations should not be protected or that law should protect him or both?