I also use the 100-400 EF zoom (v2). I have played around with mode 3 for both still and moving subjects. I don't see any advantage over mode 1 for static subjects - if anything it seemed to be slightly less effective that mode 1 but I have not conducted any systematic tests. I have occasionally used it for in-flight photos where there is unpredictable movement of the subject in both the x and y axes. I honestly can't say whether this helps in terms of subject sharpness but when the subject movement is unpredictable it helps to not have the lens 'fighting' my movement on either axis in terms of what I see in the view finder.
A couple of things intrigue me about mode 3 - perhaps someone can enlighten me on this as my understanding of the physics and engineering of these systems is minimal.
First, the lens stabilisation system must surely take a finite period to spool up to full effectiveness. The Canon Europe article seemed to suggest this is getting faster with more modern systems, but surely it is still (relatively speaking) a lot longer than the shutter activation. It takes time to start shifting a relatively heavy lump of glass. With mode 1 my experience is that it is preferable to wait a second between half press (to activate the stabilisation) and full press of the shutter button, to allow the system to operate to full effectiveness and the image to fully 'freeze' in the viewfinder. This suggests that in mode 3 either the stabilisation is 'quick and dirty' with the emphasis on speed - some stabilisation rather than optimal stabilisation - or there is a short additional lag between full press of the shutter button and the shutter firing (which if so would kind of contradict mode 3 being useful for fast action).
Second, the stabilisation motors actually start to operate when the shutter button is half pressed. Half press the shutter button to focus, and you can hear the motors whirring even though the image in the viewfinder does not freeze. This suggests there is some clutch mechanism that keeps the moving lens element in some way disconnected from the motors, and this is disengaged the moment the shutter button is fully depressed. Again, there must be some lag in this.
The other observation I would make is that while I tend to leave stabilisation on as a matter of course, mostly mode 1, I sometimes wonder whether it really adds much value in many cases. Even when sitting in one position, birds are seldom still. They preen, watch for predators, prey and competitors etc. These movements are not slow and languid, they are fast and jerky. To get sharp photos it seems to me to be good practice to keep to shutter speeds as high as possible, even if it means pushing up ISO, and I try and keep speeds above 1/500s. Looking at good photos of birds in flight, shutter speeds of over 1/2000 s often seem to be used. At these speeds, how much value is stabilisation adding (other than to help steady the image in the viewfinder)? Flash, if used, also helps freeze motion.
Simon