I am glad you find using the AF tune function on your D4 acceptable. I don't and shouldn't have to. There's no excuse for Nikon...period.
Attached are a few images that I took last night; Red fox with VR off at 1/1250 @ 6.3 - black fox with VR off at 1/800 at 6.3 my 600mm attached to the D4 on relatively slow moving foxes: AF-C 'S'. Images are unprocessed. They were about 50ft away. Focus point central and placed on their eye.