Qantas has two B747-8 freighters operated by Atlas Air for them that carry a lot of freight between Australia and the US. That does the brunt of the work for them and the 787s supplement that. UA and DL don't have any freighter capacity, so yes, they're far more dependent on passenger aircraft to carry that cargo.
QF has always had me perplexed when it comes to their freight business. Do they not actively chase a large volume of freight business? By way of comparison SQ has been running non stop with multiple frequencies per day to multiple cites in Australia even during the height of the pandemic predominately hauling freight. The same could be said of EK, QR, CX and even NZ.
It felt like QF 'scuppered' their wide bodies away for the last 18 months whilst everyone else actively chased (more) freight business to utilise their expensive assets?
Off the top of my head, QF have operated cargo flights, mostly with 330's to PVG/NRT/BKK/AKL/CHC from SYD. There have been flights from BNE to HKG/NRT/AKL, from MEL to HKG/SIN/BKK/AKL and from PER to SIN. The frequencies vary, but the SYD/PVG and MEL/HKG have been basically daily.
The 767F has been working flat out the whole time. It gets up to HKG from SYD at least 4 to 5 times a week, mostly through DRW, but CNS also sees it. They still manage to get it over the Tasman a few times a week doing SYD/AKL/CHC/SYD, when it doesn't operate, they send a 330 on the run.
The 2 Atlas 747's, seem to get supplemented with an additional 744 on at least 1 extra flight a week, sometimes 2. Their routing from the States to SYD/MEL, then outbound through China or Bangkok and China. QF have full freight rights out of China and will go load up and fly to the U.S.A. usually full. The China/U.S.A. routings would be the lucrative part of the operation.
More recently we've seen more flights to and from the U.S with the return of the 789 and they were being used mostly for their freight capacity until the recent opening of N.S.W and Victoria.