That's my answer. Well actually all three as I have top elite with Skyteam too. It's really good to be elite in more than one since your preferred alliance could be affected by a strike or something. Last year Lufthansa had a terrible strike and before that BA had the nightmare of the T5 opening. I was able to avoid both problems. Plus, with multiple options you can price shop which can be very important when you source tickets from places outside the US.

Since you live in the US, it's probably better to stick with US carriers, for the domestic upgrades if nothing else. But also you don't have to pay huge taxes and surcharges on award tickets like we do in Europe. Also, if you are only paying coach, it's best to stay with US carriers. Most international airlines, like BA as you noted, only reward business class fliers.