AKL (or SYD)-SCL vv is difficult to get first class availability. This is due to only 5 F seats on each of the 3 flights per week. Most of these are sold F not A fare class, so RTW (which books into A) and awards are extremely hard to come by. There are much more business class seats so that is easier.

Dont forget Qantas has a codeshare on this flight - so if unavailable on LA flight number check if it is available under QF code.