I think it takes some effort to figure out what works for you, with whatever lenses being used, and under whatever conditions. Lots of comments about soft or blurry video from the D4 etc. I had the same experience initially & was also very disappointed, not so now.
I continue to experiment, but I have found so far I get best results using a DX lens on my D4!
With the 10.5mm fisheye recording 1080p in 2.7crop mode I haven't found anything else to come close.
Barrel distortion is minimal for my usage needs & distance to subjects though it will become very noticeable the closer you are to whatever you have in your field of view - but I don't care about that as it is not a typical scenario for me.
Focused & set up appropriately I get the sharpest video I've seen anywhere with any other lens.
Mess up on any of the settings & things can degrade quality very easily.
Perhaps a much more expensive non-fish-eye wide angle lens might eliminate the distortion but there aren't many lenses to choose from.