I'm assuming the sandhill crane images are all severely cropped - probably to 100% or more. How far away were the sandhill cranes?
I had the same experience with my 600 f/4, TC 20E III, and D7000 at subjects approaching a half mile, but with closer subjects and controlled testing at 50-100 feet were pretty good. Stopping down and mirror up did not make any real difference. I never came up with a solution other than getting closer.
I've also done some testing with the D800E and the 600 f/4 + TC 20E III. Without the teleconverter the combination is stunningly sharp, but with the teleconverter images are slightly soft. I still am a bit reluctant to use the combination as I don't get the stunning level of sharpness I get without the teleconverter. You might not see the difference at normal viewing distances, but at 100% they are clearly soft.
As suggested, stopping down helps a little but when you are talking about extremely distant subjects and high magnification, everything must be perfect. Even then, there is a limit.