If you can get a high enough shutter speed with the 400/5.6, that's sharper than the 300/4. Both are very slightly sharper than the 100-400, but there is copy variation to consider, so you may find some 100-400s that best the 400/5.6. The 70-300L is sharper than all three.
The superteles are on a different class, for IQ and cost. My 600/4L IS II delivers substantially better IQ than my 100-400L, my 70-300L, or the 300/4L IS that I used to own.
The 300mm are only sharper than the 100-400 if you can get close enough on a 5D3 to frame as desired. For birds that is often not the case. Nothing, no matter how sharp, makes up for 100mm. At comparable focal lengths yeah the 70-300L is sharper than the 100-400 though, same for the 300 f/4 and of course 300 2.8.
Have you actually tested the 300mm f/2.8 L II against the 100-400mm at 400mm to back up that statement?