Just wanted to give an update. Firstly, I'd like to say that in my opinion, it is a pretty tricky situation to ask airline staff to process an I-551 stamp as equivalent to a Green Card, considering that I couldn't be processed by any of the usual ways. I have no ESTA visa waiver, no Green Card and no visa, just a stamp in my passport. Still, I have the relevant evidence that I'm allowed to travel back home and never doubted my right to do so, only their willingness and ability to accept it (which to me is fair enough if they were to baulk).
However, I was processed without issue at not just one but two airports in New Zealand (in accepting a seat on an earlier connecting flight, I inadvertently created a situation where I had to check in at two airports before leaving). I was always polite, kept it very simple and let them ask questions rather than over explain anything.
When I arrived in LA, I was allowed to use the line for US citizens and directed from there to the customs waiting room while they verified my stamp. That waiting room was full, but I was last in and first out. I felt a bit guilty about that!
My flight home from LA was cancelled because of a snow storm and honestly that has been the biggest issue of the whole journey.
Thanks everyone for giving info, suggestions and support in this thread! I'll be applying for naturalization asap.