Since (presumably) BA is your BEY-LHR+ carrier and you use them for transatlantic segments only, it seems logical to stick with BA, even with their awful surcharge practices.

If you were to switch to AA, under the new AA elite system you might or might not qualify for Platinum (Oneworld Sapphire, equals BA Silver) or Executive Platinum (OW Emerald - BA Gold) depending on how much you fly, but your redeemable miles would depend on how much you spend.

Another option is Qatar's FFP, but I'd spend some time looking at their offering (as well as doing the math on AA's new scheme) before switching from BA. Off the top of my head, I think given your wish list (upgrades etc.) you might well end up better off with BA than by starting new with some other airline.