Really interesting video, thanks for the link!
It's not possible to book just JFK-LAX on Qantas, although I believe you can fly a different Qantas flight on the Australia-LAX leg and connect to/from the LAX-JFK "tag".
It looks like there's a change of aircraft at LAX these days, it's a 747 from JFK to LAX, then an A380 from LAX to SYD. Back when I flew it, it was the same 747 for both legs, although after the JFK-LAX leg you had to get off while the plane was refueled, which was an absolute pain in the butt because the gate area in T4 wasn't nearly big enough.
I remember the idea of allowing American (but not Qantas) to sell the LAX-JFK leg separately by AA buying the seats from QF was floated at least a couple of times, but the FAA wasn't having it. It wouldn't have been that big of a money-maker for QF anyway, since any seat sold on LAX-JFK wouldn't be available for sale to a SYD-JFK passenger, so it'd probably have been restricted to last-minute sales. Sort of like the way, back when dinosaurs (or tri-jets, at least) roamed the earth, you used to be able to get super-cheap seats from LAS to LAX on Hawaiian (DC-10, IIRC) so they could get a little more revenue out of an otherwise-empty seat that they'd already sold between LAX and HNL.