Just change the number of travelers. If it shows availability you know at least that number are open.
Precisely! ^ Much better than BA as you can see a month at a time as well as EK, AA & JL too.
Scanning some dates this fall I found a JL F HND-LAX. I went to AS website and it showed it too. BOOKED! I also found AA SAVER F LAX-HKG; went to the AA website and it was there too. BOOKED! Seems to be quite a few options for AA at least to Asia.
AS for CX, I have had 100% success in booking awards with AS when I provide the dates, times & Flight numbers I find with QF. I just booked MEL-HKG in September and 1F & 1J CX LAX-HKG//HKG-BNE with connecting F YLW-SEA-LAX in February. This works well with a mileage run I booked from SYD with a 10 day stopover in PPT for a Tahitian cruise on Oceania. Routing is SYD-AKL-PPT-IPC-SCL-MAD-LHR-SEZ-DXB-NRT. First 4 Y last 5 J. 32,183 EQM to AS. 51,708 RDM before elite bonus.
If the OP is looking for CX J × 4, very possible, might have to split between multiple daily flights to the same city. Of course if he wants to use Marriott Travel Packages he would need to redeem 1,080,000 Marriot Points for 4 Cat 1-5 hotel packages & 480,000 AS miles; just the amount needed for 4 J return. That is a lot of Marriott Points for most but perhaps the OP is awash with them having already redeemed for Avios and a Category 8 hotel stay in Japan last December.
James