Qantas premium cabins to the states are among the hottest property out there when it comes to points, so wouldn’t surprise me to hear that you’re having issues.
Given that your points balance is with Qantas’ you’ll want to check all the available airlines, but to my mind there are two options I’d consider. If funds permit, I’d look at buying a revenue ticket in premium and then upgrading in points, or possibly look at paying a cash fare to another destination, to link up with an award ticket. For example if you could find AA awards on the Auckland - LAX flight I’d consider buying an economy ticket to meet that, or if you could look to get award flights NAN - SFO/LAX with Fiji Airways. Not the best hard product out there but I slept fine, and any business is fine by me.
I had this issue myself last year - going to Mexico and couldn’t get anything business to any US city out of Sydney, Brisbane or Melbourne. I found AA seats HKG - LAX and I had time, so I bought an economy fare SYD - DPS - HKG and took five days extra of work for a break. Spent three nights in Bali, and two more in Hong Kong to make the trip manageable, relaxed, and got me a long-haul business seat into the USA.
I realise it may not work for everyone, but flexibility is the best thing to help you in your search. If you need the most direct, most desirable flights, there’s a good chance you’re out of luck cause everyone else wants them as well. Whereas if you’re willing to be flexible, do something a little quirky that most people wouldn’t want to do, you increase the chances of finding a seat.
Best of luck!