I think with the right itinerary you could get to Emerald with any of BA, QR, AA, or AS quite easily. There are other threads for xONEx optimization, but since you're choosing the FF plan based on earnings and status, throw up a tentative itinerary you're working on and we can let you know the feasibility. BA and QR use a block based tier point system so maximizing long hauls wouldn't be the primary goal, while AA you'd want to maximize long hauls with specific partners and avoid certain others, with AS they all credit pretty nicely but you'd want to try for some AS segments to bring down the number you'd have to do in mileage runs.
When you're checking for earnings rates, you'll be looking at D fares (I'm assuming DONEx here, but don't remember if you specified, I don't think you can get to Emerald on an economy ticket alone).
Wheretocredit.com gives you some idea as well, just change the airline at the top but you'll be starting out at base level and moving up through the tiers quite quickly as you fly where bonuses will start kicking in on some airlines. Bonus earnings won't matter for status on QR or BA but certainly will if you go with AA.
When considering BA, keep in mind BA, IB, and QR Avios are interchangeable so when one has a better award chart or availability than the other, you can move the Avios instantly between them, presumably AY will be the same next year though at the moment their award chart and program isn't very appealing, have to wait and see if that changes next year or not. EI (Aer Lingus) also participates though I've personally never found any use for their awards, but from the US to Europe you may very well find it useful and their business class seems quite acceptable (however that's long haul only, intra-Europe is economy only).
QR has the same 4 segment requirement as BA, but I wouldn't dismiss QR Platinum (OW Emerald) if your travel will continue to take you via DOH. Platinums get access to the First Class Al Safwa lounge when on QR business tickets (a DONEx would count if you were QR Platinum already). It's extremely easy to put 4x QR segments on a DONEx to meet that requirement and they have much lower fuel surcharges than BA.