I have the 100-400L which is about the same size/weight of the 2.8, so I know what it's like carrying a tree stump around all day. When you need to, you need to. But often I head out with my Think Tank Retro 5 stuffed with my 5Dmkll + SP24-70VC + either 17-40L or 70-200L with no particular photographic mission in mind. I just like to have my stuff. I don't think I'm likely to do that with the 2.8L. So I'm screwed in that I think I will "need" to keep my f/4 for such trips as well as when "traveling light."
On a positive note, I just got the shipping label from Canon so my new heavy toy should ship tomorrow, yipee!
I find the 100-400 is (a lot) easier to carry, esp with a tele converter! I very often have the 1.4TC on the 70-200 so here are the weights:
70-200 IS II : 1490 grams
100-400 : 1380 grams
70-200 IS II + 1.4 II (=98-280 f/4) : 1710 grams
Also, the 70-200 seems a lot more front-heavy than the 100-400, this gets worse obviously if you add the TC. The 100-400 gets long when you point it at (fast) things far-away which makes it easier to 'aim'.
This is why I prefer the 100-400 on sunny days when I need the extra reach, but I will live with the extra weight of the 70-200x1.4 when I need f/4
5D3, 5D2, Sony α6000, G16 | SY14 f/2.8, Ʃ20 f/1.8, 24 f/2.8, 35 f/2, Ʃ35 f/1.4A, 50 f/1.8 I, Ʃ50 f/1.4 EX, 100L Macro, 17-40L, 24-105L, 70-200 f/2.8L IS II, 1.4x II, 2.0x III, 70-300L | E-mount: SY12 f/2, 30 f/2.8 EX DN, 16-70 ZA OSS, 55-210 OSS, Metabones SB | FT-QL, AE-1P | FD(n) & FL lenses