I'm not a sports shooter, but I do have both of those lenses. I have used the 70-200 for indoor basketball, but even at f/2.8 and 6400 ISO I could only get 1/250 for my shutter speed, so I'd say that f/2.8 may not cut it for shooting sports indoors. The MK I version, which I have, is also not razor sharp when used wide open. If you want to shoot indoors then you might need to look at lenses such as the 200/2L, 135/2L and 85/1.8, or you may need to set up off camera strobes.
Outdoors I find my 100-400 to be pretty good for motorsport, where as often as not I am trying to slow the shutter and really do not need fast apertures. I also use it for shooting BIF, with some success, but the AF is not the fastest and overall I don't think you'll find a pro shooter using a lens like the 100-400 for any fast action sports, or sports full stop.
More typically I think you'll find the pros will be using lenses such as the 300/2.8, 400/2.8, 500/4 and 600/4, and perhaps have a brace of teleconverters to round out their range. Such lenses will give them the focusing speed, shutter speeds, IQ and weatherproofing that they need to shoot in all conditions. A leaky, slow zoom like the 100-400 is not going to deliver the goods when shooting under floodlights on a rainy November night.
So, to answer your question, I'd say that to do the job well your proposed lenses will probably not deliver results a pro would be proud of. Whether they would be good enough for your needs I do not know. If you can be more specific about the types of sport, whether indoors or out and what sort of weather and lighting you anticipate then people may be able to provide more helpful answers.