I am having the same issue with my 70-200 2.8 IS II on my 7D mark II. At 200 @2.8 it is quite soft until I stop it down to f/4 and beyond. Used to be super sharp on my 6D.
If I do the test, I will pick a good light day to shoot the nearby object like rock or tree, or a tripod. That way you avoid all the different factors that may affect your result. Shooting moon is not good test of your lens, IMHO.
Lots of people use the 70-200 mm to take portrait too. That is another way to test your lens.
The measured performance of the lens at 200mm shows it to be extremely sharp. It is not as sharp as it is at wide angles, but still extremely sharp. Some of the issue with the test results is just distance. Even 100 ft can reduce apparent sharpness. Getting sharp moon photos is a art. People staying up until 2 or 3 in the morning in very cold and clear weather over several nights with extreme care to eliminate vibration. Even a good tripod transmits vibration, so hang weights and sandbag legs on solid ground, no porch or deck will do. Get away from city lights, a long ways away.