If I had the cash, I'd use the 70-200 f/2.8 sans TC when the wide aperture is needed, and the 70-300 when more range, less weight/size was needed and aperature less important... They complement each other well.
The 100-400 I think you will always be stuck with a heavier+longer lens even if redesigned, and it doesn't differentiate itself as much from the 70-200 as the 70-300 does (aside from the controversial push-pull zoom).
Others have pointed this out as well but the lens differentiates itself by having twice the reach as the 70-200 and 33% more reach than the 70-300. And the 70-300 can't mount Canon TC's (not sure what quality is with alternatives - might not be bad) while the 100-400 can and still AF on a 5d3 and still provide decent quality.
That is true, but the 100-400 also approaches the 70-200+2xTC in weight and size. It is also less sharp than both the 70-200L and 70-300L, while being larger and heavier than the 70-300L.
And the push-pull isn't really controversial - you either like it or you don't. If you like it you are happy and don't care what the rest of the world thinks about it. If you don't like it you don't get the lens. I personally like it.
You kind of just defined controversial
It all comes back to what you are doing with it. If you need the 400mm reach for your application, the 70-300L will leave you dissatisfied even thought it is a great lens.
Maybe not. Would you rather have a 70-200mm f/4L or a 70-300mm f/4-5.6 non-L? The latter has 100mm more reach but is not as sharp. The same could be said about the 100-400L vs 70-300L. Reach is not always better if it means sacrificing image quality.
If you only occasionally need the reach perhaps you can get by with 70-200/2.8 plus some TC's but that involves compromises you have to be OK with (I agree that ergonomics of 2x on 70-200/2.8 are not great, uncomfortable to carry).
Or you could buy a 70-300L, have something of much more compact size and lesser weight than either of those two. And, when you need the 400mm reach, you crop in post. The point is instead of another monster sized lens you have something a bit more managable that is also sharper than the 100-400L.
The 100-400 is a compromise lens. You can find other lenses that will beat it at each and every focal length but there aren't many options that provide the range and flexibility in one lens. That's why it continues to sell well. Lenses are just tools and just collecting all the best ones is not the recipe for happiness or success in your photography. You have to get the ones that meet your needs.
This is true, everyone has their own opinions and preference, of course