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Conceding that 2 cameras and 2 lenses are the best way to go, what do you do if you want to consider just one camera and one lens?
The answer then has to be the 70-200 f/2.8 with the 1.4 extender. Doing it that way, there will be parts of the field in outdoor sports that will be beyond your reach, but (for most field sports except American football) they will also be the parts most likely to have line-of-sight obstructions between you and the action. No matter where you stand, the closer the action gets, the less likely there is to be someone irrelevant blocking your shot. Which makes a long lens like the 400mm kind of limiting, because it's going to be at it's best on the parts of the field where someone distant is more likely to be obstructed by a ref or a stray player who isn't part of the action. Even the 300mm is often going to be too long for the closer parts of the field where you are most likely to get unobstructed shots of the action.
But the zoom gets a big piece of the best of both worlds. With the extender on, if the action is coming right up the sideline, you can step back a bit and catch it at 100mm. Or you can zoom out to 280 to cover at least a third of a soccer field, maybe a bit more, when the action is more distant. When you go indoors, you take the extender off, and you're set for basketball, hockey, what have you, with f/2.8.
It's not the way to go if you intend big-time sports coverage at the highest level—there you need two cameras plus a long prime and a zoom. But for most people, most of the time, the zoom with the extender will provide many great keepers from every event.
By the way, using my 70-200 f/2.8 IS II with the 1.4 extender II, and my 5D III to shoot soccer, lacrosse, track and field, and basketball, the autofocus does not seem to be challenged in the least. At track meets I count on shooting from the end of the pit, and getting multiple sharp in-the-air shots of each long jumper coming straight at the camera.