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.
+1 My 500mm f4L II has much better IQ than my 400mm f5.6L. The 400mm f5.6L does play well with the 1.4x III on the 5D3 and with the Kenko DGX 300 1.4x on a Rebel. The 100-400mm and extenders are much more hit and miss.