All you need to do to increase your keeper rate with the 6D, is set "second image" to more accuracy rather than speed (in the autofocus menu). It will slow the frame rate down, but so what? Basically all the frames will be razor sharp. I admit I'd like to have really fast frame rate and accuracy sometimes, but most of the time I don't need a lot of fps.
If you need more responsiveness on the 6D, set "C.Fn II:Autofocus" under menu #1, all the way to the right, "responsive". I'm also using "accel/dec" tracking all the way to the right. First image priority is set to the middle, but second image priority is set to "focus", again all the way to the right.
With these settings, and center point only selected, I was able to track my cat running at me in very low light with only an f/4 lens (ISO 25,600). My shutter speed was too slow (1/100), but in slightly better light, it would have nailed him.
Birds in flight are very difficult, but I've snapped quite a few. I congratulate those of you who do this! The smaller faster birds, are the difficult ones. Larger birds are easy. I've even snapped some bats in flight at dusk, with my 135 f/2...but the animal is not pleasant to look at. Of course it only works with sky as the background, unless they happen to come out early enough.