I initially bought both the 1.4 III TC and the 2X III TC to go with my 70-200mm f/2.8 IS II and 70-200mm f/4 IS. The 1.4X TC definitely gets more usage, but the 2X has hardly been used since I threw in the towel and bought the 400mm f/5.6. BIF with a 6D and the 400/5.6 has worked out well for me. Sometimes I use a 60D if I am shooting small birds.
My problem with the 70-200 f/2.8 IS II + 2X TC III was static shots were fine until a raptor took off and I would lose focus in the sky and I would have no image at all. I could never reacquire focus unless I focused on a tree or the ground. By then the subject was long gone. Focus speed is certainly slow with the 2X TC compared to the f/2.8 zoom alone, which is very fast. Of course it depends on where the lens was initially focused in the first place how fast it reacquires focus.
The 400mm f/5.6, however, has been great in normal daylight and keeping the speed up around 1/1000. Hindsight I would not have bought the 2X III TC for the lenses I have. From a purely image quality standpoint, I have no complaints about the 2X TC, at least the center portion of the image with the 2.8 zoom. If I shoot the same static shot 5 times with the 2X TC on the 70-200 2.8 IS II, and refocus on each shot, and then shoot the same subject 5 times with the 400mm f/5.6, I get more focus variation from shot to shot than between the lenses. This is with a subject about 20 feet away, and resting the lens on a bean bag.