The 70-200/4 IS would be my choice... used with a 1.4X teleconverter on occasion. It won't give you quite as much reach as the 100-400, but has excellent IS and image quality, "workhorse" build quality... should be long enough for most field sports... and is compact, easily handheld. There will be times it's good to have f4 instead of f5.6, too.
All the 70-200s are traditional "two ring" zooms... one ring to zoom, the other to focus manually. The 100-400 is a push-pull zoom... one ring does both and you slide it inward or outward to zoom. I'm not a fan of this type zoom, I find them harder to get a steady shot. Some folks like them and they are fast handling (although with modern AF cameras it's just about as fast to use only the zoom ring on a two-ring zoom).
I should qualify the above that I use a couple different 300mm and a 500mm prime, too, all with and without 1.4X and 2X teleconverters.
And, don't be tempted by a 2X teleconverter... perhaps you already know it will effect AF when used on an f4 lens. I also find the loss of IQ to be too great with a 2X on a zoom. 1.4X is pretty darned good, though. And modern 2X work quite well on many prime teles.
Shooting sports for many years, I couldn't disagree more with some comments that IS is of no importance for sports shooters. There are quite a few thousand other pros using it, too, who will strongly disagree. Every other camera manufacturer has had to scramble to catch up with Canon's IS lenses... Which utterly dominated every major sports event for years where their only real advantage was IS. Now that others are offering VR, OS, VC etc., Canon lenses are no longer as dominant.
Sure. Non-IS sports shooting certainly is "doable"... We all survived somehow for many years without it, after all. Just plan to predominently go for one style of sports shot... Frozen action done with a high shutter speed. Actually, IS can help with those too, and is useful for a variety of other kinds of shots... panning, dragging the shutter to get deliberate subject blur, etc. In the end, I'd much, much rather have IS and not need it, than not have it when I do need it.
There is acutally no reason to ever turn off IS on many of the lenses, unless you are trying to save a small amount of battery power. Even if you are going for a panned, blurred background, you actually can leave turned on but should use the correct single-axis mode.
There are a few lenses where you should manually turn it off when the lens is on a tripod, locked down: 100-400, 300/4, 28-135 are several of these and there are some more. It's not necessary to turn off even on a tripod with any of the 70-200, 300/2.8, 400/2.8, etc.... These lenses sense "lack of movement" and turn it off themselves when not needed.