Thai has stopped service, now nonstop from BKK to AKL.
EK & QF have much tighter pitch than TG or NZ. Not sure about the others DJ, LA, AR, SJ, BI, etc.
Between ADL and AKL only nonstop options are NZ and QF.
Have you thought about shuffling the itin around? If fly UA into MEL then could do QF or DJ MEL-ADL-SYD-BNE fairly cheaply, then NZ BNE-AKL on 747 (ie longhaul product) and AKL-SYD or MEL then UA back to usa.