The 200 f/2.8L is a really nice lens. However, it AF's way slower than the 70-200L and offers a lot less flexibility, which is why I don't think it'll help him much for sports, whereas the zoom, and yes quite expensive, will. It'll be a tough choice. Two bodies/two primes or a single body and the zoom.
The EF 200mm f/2.8L II USM prime lens AF it's plenty fast for me. Fast enough for birds in flight. Now the last time I checked humans are slower than birds. My kid is crazy fast maybe a track star one day, but I have no issues capturing him with the EF 200mm f/2.8L II USM prime lens either. It sounds like the OP enjoys prime lenses. This one is affordable and can do the job, what is not to like?
If he is stuck between these two amazing lenses then why not rent both and see which one is better for the type and style shooter he is now and wants to become?
That's great. I'm sure it does very well for BIF and kids running outside. Try it in a dark gym. Then it isn't. That's all I was trying to say. I too am not suggesting for him to buy a lens that costs +2K. I'm just objectively comparing the lenses since I've done it myself and chose the zoom, because of what I already said, AF and flexibility, not to mention IQ and IS. Expensive, yes. Maybe the 200 prime is the best for him, who knows.
Good idea to rent each.
Going to try to keep this post nice. After all it's a holiday.
I'm not sure I understand your post regarding a dark gym. Both lenses wide open are f2.8? Not to worry, this will be my last post to this discussion. Please don't hear me wrong the 70-200 is a GREAT lens. Zoom lenses can be a wonderful solution. I think sometimes it is taken too far regarding it being "the only real solution". You don't have to take anyones word for as they are sharper prime lenses than the 70-200, even Canon's own mtf charts will tell you that, but the real test is when you have the lens in your own hands and compare the result(s) for yourself. Depending on the indoor sport you maybe able to get the shots you need with 85mm f1.8, 135mmL f2.0 or even one I don't think I've seen posted yet: 200mmL f2.0 or just take the f2.8 telephoto zoom and crank up the ISO and fix in post no big deal. I've taken a portrait on a crop body using the 300mmL f2.8 (+ extension tub to reduce MFD) lens and guess what, they are pretty good. Now is that "traditional" no, but who cares? If it fits your style and purpose then no harm right? The point is that no one lens has to be everything to everybody. From the almost rude nature of a few of the post on this thread appears that this logic maybe lost on those few. Maybe sometimes the best lens is the one you already have on your camera? Maybe its not just your gear that makes a good or even a great image...
Well, I understand the OP does not wish to rent, they are also good retailers that will let you return a lens within 30 days of purchase. So perhaps try a few of them and return the ones that don't make the cut?
jdramirez: I hope the lens you do select in the end works well for you. To be honest it sounds like if you do choose from the lenses already posted that you really can't go wrong
Good Luck and Happy Shooting!