More complicated answer - sort of.

I know you can sign up for Asia Miles and search for award seats on CX and that what you find matches up pretty well with what you'll get by calling AA for a OW or Partner award.

I can't speak to JL's system.

But while it's true that the availability for their own FF's is expanded, it's certainly true that there is a major connection, and I've found CX's online availability tool to be a good guide for AA miles based awards. Try the OW forum as well, there's a lot of discussion of availability tools in a few places on FT.