First off, don't use Air New Zealand as the benchmark. Aside from the fact that they're not hiring external applicants at the moment, the chances of a non New Zealand national joining from overseas are slim to none unless coming in as a Sim Instructor. Even expats struggle to join the National Carrier. Not sure what things are like over in the States, but it might be akin to me trying to apply direct to AA/United/Delta... or some other Airline that almost entirely hires locals only? It's not impossible, just highly unlikely.
Not true, the requirement is to be able to live and work in New Zealand. If you have residency and a NZ CPL then you are as much in the mix as any "National". I know personally a number of Aussies, Poms, Scots, Irish, Middle Eastern and South African nationals working in the Jet Fleet. The TP fleets have a few Japanese, Korean and Chinese nationals as well.