Just DL being a travel agent and booking an AU domestic airline. Nothing unusual. Until recently UA would book, QF, JQ and/or VA for AU domestic flights.

PER NYC via Asia is viable option. PER can be difficult
Getting without many transits is hopeful. Would expect 3 or 4.

USA -SYD-USA return, SYD-AKL-SYD and SYD-PER-SYD may be the way to go, if price is a driving factor. For the client expect your ff miles are secondary.
Would expect prices for Aug 2023 are currently high.

Air NZ is Star Alliance, VS is a ff partner ---> https://www.wheretocredit.com/air-new-zealand
UA is now a ff partner of Virgin Australia VA.
Be wary of orphan ff programs. FF miles can expire before getting enough to use.
-->Rex Australia and Delta Air Lines Announce Partnership Will be of no use for your trip.

Map 1--->A map from the Great Circle Mapper - Great Circle Mapper
Map 2---> A map from the Great Circle Mapper - Great Circle Mapper


Airlines that fly USA/Canada to/from Australia
Following the pandemic, routes are being re-established and some new routes. Schedules can change. Route restart dates subject to change.

• Air Canada AC YVR – SYD BNE. [Star Alliance] .
  
• Air New Zealand NZ LAX SFO IAH JFK YVR via AKL. [Star Alliance]
  
• Air Tahiti Nui TN LAX – via PPT Tahiti & AKL. Last segments on QF codeshares. (AA partner)
  
• American Airlines AA LAX-SYD due to restart 29 Oct 2022 [OneWorld Alliance]
  
• Delta DL LAX – SYD. [Skyteam Alliance]
  
• Fiji Airways FJ LAX SFO HNL – NAN - BNE SYD MEL ADL. (OneWorld Alliance connect. QF AA & AS partner)
  
• Hawaiian HA HNL – SYD. Pre CV19 also used to fly to BNE (AA & VA partner)
  
• Jetstar JQ HNL – SYD. Pre CV19 also used to fly to/from BNE & MEL.
  
• Qantas QF SYD - LAX DFW HNL YVR*; BNE - LAX ; MEL- LAX. {* seasonal}. MEL-DFW due to start 02 Dec 2022 (new route). SYD – SFO due to restart 30 Oct 2022. SYD-AKL-JFK planned from June 2023. Pre CV19 also had SYD-SFO & SYD-JFK(via LAX). [OneWorld Alliance]
  
• United UA MEL-SFO, SYD-LAX SFO. BNE-SFO from 31 Oct 2022 (new route). MEL-LAX and SYD-IAH due to restart 30 Oct 2022. [Star Alliance]. (VA partner)
  
• and others via Asia & Middle East
  

Airlines that fly USA/Canada to/from New Zealand (not from all airports to all airports).
Following the pandemic routes are being re-established and with some new routes. Schedules can change. Route restart dates subject to change.

• American Airlines AA . DFW-AKL due to start 29 Oct 2022 to 25 Mar 2023(new route) Pre CV19 AA flew LAX-AKL and was due to fly LAX-CHC 3 days a week from Oct 2020. [Oneworld Alliance]
  
• Air Canada AC YVR – AKL Due to restart 12 Nov 2022 [Star Alliance]
  
• Air New Zealand NZ LAX SFO IAH JFK YVR – AKL. JFK started 17 Sep 2022. ORD from late Oct 2022 [Star Alliance]
  
• Air Tahiti Nui TN LAX – PPT - AKL (AA partner)
  
• Fiji Airways FJ LAX SFO HNL – NAN - AKL WLG CHC. (QF AA & AS partner).
  
• Hawaiian HA HNL – AKL (AA & VA partner)
  
• Qantas QF SYD-AKL-JFK planned from June 2023. LAX SFO DFW JFK HNL YVR* via SYD BNE & MEL. {* seasonal} [Oneworld Alliance])
  
• United UA SFO – AKL Restart from 01 Oct 2022 [Star Alliance]
  
• And others via Australia, Asia & Middle East