You'll have to compare your own earn/burn rates on the various plans, but in general AA is 50%-150% more generous (i.e. you get roughly half as much free travel when crediting to the other OW programs). It varies a bit by how you earn and burn, also the non-AA programs have some step discontinuities (so it matters which side of the step you are on).

Any non-AA Emerald (or Sapphire for that matter) has access to the AC when flying AA domestically (that is why I have QF Emerald status along with EXP). The primary reasons to pick non-AA are non-mileage benefits.

CX Diamond: can purchase discounted fares even when sold out, as long as you otherwise qualify (e.g. 21 day advance purchase can be bought even if the flight is sold out with only Y inventory). Sporadic (and sometimes frequent) undocumented benefits, but this might vary with how profitable you are to CX (they definitely track profitability by individual pax, at least in some cases; most other airlines do not). CX upgrades can be bought only using Asiamiles (CX plan) and not using AA or BAEC.

BA Gold: "open doors" access to BA lounges even when not flying Oneworld (QF Platinum also has this feature, but far fewer lounges). Earning BA status is based on length of flight, with some magic thresholds. So flight over 2000 miles earns much more for status than one of 1900 miles. Making certain routes lucrative for earning BA status (particularly AA YUP fares which count as F fares in the BAEC plan, though this loophole may be closing soon). Makes it easiest to earn BA Gold status (compared to CX or AA), but BAEC miles are worth the least of the 3 plans.

So the no-brainer is to stick with AA, and you probably will get the most bang for the buck. Or spend a couple of hours and map out what you would earn on CX and BA plans, and see what awards you can use it for. You earn fewer miles on CX but the miles are worth more (in some cases). At first I thought the CX plan was by far the worst, but I now think it is the best plan if you travel a lot on CX, can make Diamond for a few years, and factor in the unofficial (and unexpected) benefits. The "purchase discounted fares when sold out" benefit is a huge benefit (but only on CX flights).