I was in your position a few months back. Infact I actually owned both side by side for a while and did some tests. (70-200 f2.8 mk.ii + 2x mk.iii vs. 100-400mm).
I agree there was slightly faster focus with the 100-400, but the f2.8 +Tc was not bad at all. On my 5D3, the AF is still pretty fast , I am able to shoot sports with not a lot of issues with this combo.
Second, the contrast was abit better with the 100-400, but the gradients in the TC combo were better and let me see more details overall. Also the contrast can be tweaked in PP so this was not a huge issue for me.
Third; Sharpness was a toss, but perhaps the 100-400 was slightly better... but not by much. the 2x mk.iii is a huge improvement over the 2x mk.ii so talk to people that have used the mk.iii TC for a more relevant comparison. (unless you plan to buy a 2x ii in which case IQ will take a definite hit)
Fourth: the 100-400 is a large lens... takes up a lot of space in my camera bag, I can store 3 lenses vertically for that 1 slot that the 100-400 takes.
Fifth: Weight. No comparison here.
Sixth: you can use the TC with other lenses also
Lastly you will save about $1000 and not lose a whole lot in IQ, in fact youmight need to pixel peep to see a difference. I am assuming you do not have deep pockets otherwise you'd be getting a 400mm f/2.8 ii.
The dedicated lens will always be better but the question is, is it a material difference?
I sold my 100-400 a few weeks back, and have little desire to own one again while I have the 70-200 mk.ii plus TC combo working.