I can't speak for the JFK route as I haven't redeemed on that in ages, but broadly CX has severely cutback partner inventory availability in favor of its own program (Cathay / Asia Miles). Skip searching AA and BA for now and just use CX. Then, if you find dates you like you can go backwards and look at the partners and see if available.

At least regionally, where I've redeemed lately, CX has tons of award inventory available via its own program. (Although I've been advised by forum members here that's a dumb use of my points.)