I don't have a 100-400 to compare to my 300 f4, but the resolution tool over at the TheDigitalPicture tool show the 300 to be much sharper than the 100-400 at 5.6, center mid and corner. And the 70-200 2.8 looks really sad at 200, and 280, and 400 (with teleconverters) And the 300 still looks slightly better at 420mm (with a 1.4X) than the 100-400 at 400 (both at 5.6)
I agree that the 300/4 bare is sharper than the 100-400mm at either 300mm of 400mm. But I think the 100-400mm @ 400mm is slightly sharper in the center and mid-frame than the 300/4 + 1.4x II/III, and equivalent in the corners (and the corners aren't relevant if you're using a 1.6x body). Here's the link
for those that want to compare for themselves.
Personally, I have a 100-400mm, had a 300/4 (and still have a 1.4x II), and I found the 100-400mm to be better at 400mm than the 300/4 + TC.
Of the 70-200mm series of lenses, only the 70-200mm f/2.8L IS II holds up decently with teleconverters. But IMO, a native lens will almost always beat a lens + TC, so if you need a focal length, get it if you can, and use extenders sparingly. The exception is lenses in a dramatically different price class, such as the new 300/2.8 II + 1.4x III, which bests the 100-400mm @ 400mm. But I will point out that design age is playing a factor there, too - the 100-400mm @ 400mm is still slightly better then the 300/2.8 MkI + 1.4x II.