After having used the Sigma 150-600 C for some time now, I think I can confidently conclude that:
1) I have no regrets selling my previous 100-400L for the Sigma.
2) If money's no issue, I'd like to get the 100-400L II, as well. These two lenses aren't apple to apple comparison, really.
3) 100-400L is superior for air shows and probably BIF and other action shots than the Sigma.
4) The Sigma at 600mm and 100-400L (original) at 400mm are about the same sharpness-wise, perhaps even a slight edge to the Sigma.
5) When the subject is very far, however, I believe 100-400L does a better job given the same atmospheric influence.
6) I can rely on 100-400L at pretty much all circumstances; with the Sigma, I need to be lot more selective.
Regarding number 3 I would strongly disagree with you that the 100-400 would be better for airshows, especially here in the UK. Thanks to two incidents, the 1952 Farnborough crash, and the crash a few years ago at Shoreham, where in both cases there were fatalities on the ground, we have significantly grater separation distances than are usual at US airshows. Even shooting on an APS-C sensor @ 600mm I often still need to add a 50% crop, giving the same FoV as shooting at 900mm. That is unless I just want to shoot the aircraft from directly side on. I much prefer to get a mix of angles in my shots.
AS far as 4 is concerned I used to rent the original 100-400 for airshows before I got my Sigma. Compared to any of the Canon lenses I rented I find my Sigma significantly better at 600mm than the 100-400 I at 400mm. Yes I would really like at 100-400 II too, but given my above comment I'd still be picking the 150-600 for my airshow needs. As you say the two lenses aren't totally apples to apples as far as comparison goes.
Finally 5. Over many years of using high magnification optics, not just camera lenses, but also spotting scopes and telescopic sights, I have come to the conclusion that in general the better the quality of the optics the more they will show up mirage effects. I shoot both Bench rest and F-Class rifle where magnifications usually start at around 20× and goe up to 60× or even 80× magnification. When I did my NRA GB rifle coach course one of the instructors was the GB Palma Long Range Team's chief wind coach. His Zeiss spotting scope was close in price to a Canon 600L. On a cold March day at under 10°C that spotting scope showed that there was some mirage visible. Using optics that still cost in the low thousands you couldn't see any mirage at all. That was an extreme, but I still see it across the range of scopes I own. One of the reasons to use such high magnifications is so that you do actually see the mirage effects, since they are the best indication of wind strength/direction. This is though a comparison that must be made at the same focal length/magnification. Comparing 400mm with 600mm the 400mm will always win, even if it's just zoomin the same lens.