The problem with using the 1.4x TC is that the 70-200 becomes a 100-280 f/4...limited a bit at each end, certainly not any sharper (and I say less sharp) than the bare 70-300L. If he's using it on a crop camera like the 70D, it's really more like 140mm at the wide end. I personally find that when I am using a zoom, I use it all over the range, and not just at the long end. It's really just a pricey lens that is not all that useful as a walkaround, it's more of a status symbol. You want to feel like you're a pro photo journalist, so you buy what they use. But they aren't shooting wildlife or anything at an extreme distance. Rather they are trying to get as close as they can, and when they can't get close enough, they can go out to 200mm. It's really better for close range portraiture, than for wildlife, in my opinion. Another plus for the 70-300L, is it is actually f/5 up to 220mm, so it's not that much slower than f/4. It's also f/4 up to 100mm, and f/4.5 up to 150 or 160mm. It autofocuses pretty fast, I doubt it is much slower, if any, than the 70-200 f/2.8 ii, with a 1.4x iii on it. The 70-200 f/4 (non-IS) that I had for 4 years, autofocused a bit faster, but it wasn't a huge difference. The body you're using it on makes more difference, regarding the AF speed.
The 70-200L II can also take a 2x and still AF. You can stick a Canon 1.4x III on the 70-300L but the OP won't be able to AF (f/8 on the 70D) and he won't be able to use much of the shorter range either because the rear element of the 70-300L interferes with the TC. The kenko TC is an option, but then you can't AFMA else you might lock it up. The 70-200L II is a lens of choice if weight and cost are not issues. It's a superlative portrait lens and is easier to use indoors because it is f/2.8 and gets you to comparable IQ to the 70-300L near 300mm while being a stop faster, and can get you to 400mm with slightly worse IQ (albeit slower AF) than the 100-400.
I used both the 70-200L II and the 70-300L on a 7D, 5D II and a 5D III. Servo AF is much better with the 70-200L II (same body). The smaller max aperture of the 70-300L also causes it to fail to lock in one shot AF as well. Tried taking a shot of a wet seal on bright sunny day, and the AF would not lock on the seal with the 70-300L. Had to lock on something at the same distance instead. The 70-200L II can use the more sensitive AF baselines, and it mattered in that case.