Basically what Demonoid said.
Economy fares are not even worth exploring on AMEX Plat/Cent because they simply do not have the really cheap sale deals that airlines do. Note that you should get a complementary CX Gold membership with a plat charge card which should have a guaranteed economy ticket benefit (and you can specify cheap fare classes such as V), at least when you are flying on CX. I think you do need to give them some minimum notice to use this benefit. As a diamond myself, I think the guarantee is up to 24 hours before departure so I'd imagine gold has a slightly longer notice period.
However business and first class fares can be fairly competitive with AMEX travel. Since I have to call AMEX to book the hotels anyway (FHR), I ask my executive if he can see any special airfare deals to my destination and generally, I find the fares fairly competitive.