The 400/5.6 is an excellent lens, no question. For BIF, you'll be shooting at shutter speeds high enough that IS isn't needed.
For me, it's a question of compromise, exactly as you say - flexibility, IS, and convenience. With a 70-300L, a 400mm prime makes sense - assuming you'll have time to switch lenses. Not sure, but I expect Greg Downing may have a body for each of those primes.
The 100-400 does fine for focusing on BIF, IMO. Not a great shot, but as an example here's one I took yesterday. The camera was hanging from my BR strap when I saw this red tail take off for a short flight from one tree to another, so this is a snap-shot - grab camera, raise to eye, focus and shoot. It grabbed onto the bird desipte the trees in the background.
I believe Greg uses two bodies, as do I. He feels strongly that the 400/600 combo is the most versatile set-up for BIF (he replaced his 300 f/2.8 with the 400 f/5.6 to reduce weight). Interestingly, when I asked him if I could only afford one lens for BIF, he was adamant: get the 300 f/2.8 + a 2x extender.
I too have friends who think the 100-400 is a great, versatile, fast-enough-focusing lens. But since I already own a 70-300L, which I love, the 400 prime seems to be a better fit.
A little off topic, but since I know you're using a 1DX, what AI Servo focus parameters are you set to for walking around, so that when a hawk jumps up in front of you, you're ready?