"But without rebooking those QF flights he wouldn't make it to platinum at all (nor would he make it to the 20,000 miles required to redeem a reward). "

Yes he would, though he wouldnt have enough points for an award.

The AA flights

AA LHR-MIA 4437
AA MIA-CUN 531
AA CUN-MIA 531
AA MIA-DCA 920
AA JFK-SEA 2421
AA PDX-DFW 1616

would attain the 10,000 qpoints required to attain platinum status. Where travel is on AA, L earns 1 qpoint per mile, so the above flights would earn him 10456 qpoints on their own and so complete the challenge

If the remaining flights

AA DFW-LAX 1235
QF LAX-SYD 7488
QF SYD-PER 2041

were booked onto AA , then they would earn 21528 miles too, which added to the 10,456 plus 1616 bonus for completing platinum would give him 33600 miles

If the

QF PER-SIN 2421
QF SIN-LHR 6752

are converted to BA, then they would earn 2294 miles and qpoints and the challenge would be completed 1 flight earlier giving another 2421 miles giving a total earning of 38315 miles which is almost enough for 2 flights within australia or between australia and new zealand

To get a few more points ( plus comfort ), he could pay the supplement for WT+ on the SIN_LHR flight which is , iirc, USD300. That way he gets a mre comfortable flight , will complete the platinum challenge just on the SIN-LHR flight ( T class on BA will earn 1.5 qpoints per mile ) plus 110% mileage on the flight. He would then also have lounge access at LHR before heading off to MIA. Earning of miles to LHR would be 14120 (rather than 2294). With the bonus, the total mileage earned on the trip would be 56560.

Dave