Maybe something wrong with your 70-200.
...or maybe with your 100-400.
The 70-200 II + 2xIII is sharp, it's that the 100-400 is a little sharper @ 400mm, and the 400/5.6 is a little sharper still. As for a 'bad copy', TDP tested three copies of the 70-200 II, and none of them with the 2xIII beat the 100-400 @ 400mm.
I like Artie, but the thing to keep in mind is that his blog is there to earn money for him ("Please remember to use our links") and the ~$2500 70-200 II + TC combo will earn him more than the other options.
It seems like several of his posts are actually comparing the 70-200 II + 2xIII to the 400/5.6 prime, and favoring the former for IQ. I'd have to declare shenanigans on that...
I can no longer trust the TDP ISO charts, since my experience with certain lenses is opposite what his charts say. Example was my Canon 200mm f/1.8L lens. Mine was super duper sharp, sharper than my 200mm f/2L IS was. But his ISO charts indicate that the sharpness was very bad (just look at em). There is sometimes sample variations and so... he's also using a 1Ds III, which well, I don't think can stand with the 5D III or 1Dx sensors.
My 100-400mm was indeed sharp, but the AF was not nearly as fast as the 70-200mm f/2.8L IS II + extenders. Sharpness wise, I think they are at very least on par. So why bother keeping another big lens around. A similar situation was when I compared my 600mm II + 1.4x with my (now sold) 800mm bare. Even the 600mm II + 2x was sharper than the 800mm bare and I thought my 800mm was a good copy.
I had a 400mm f/5.6 also. It was sharper than the 100-400mm, but sharpness is not everything. The 100-400mm and the 70-200mm have much better color saturation and micro contrast. Simply put, the 70-200mm f/2.8L II is the sharpest telephoto lens by Canon. It has fluorite elements while the other two do not. Its AF is the fastest of any zoom. And it has the latest gen IS. Why not take advantage of that tech slap a 2x III on it and be done.
70-200mmII + 2xIII @ 278mm . 1/1600 s . f/5.6 . ISO 400 . +2/3 EV