As mentioned above, you should check out the lensrentals blog article with regards to focus accuracy.
It says 5D III in AF mode yields a more precise focus with the 70-300 L than the 70-200 II L (almost half the value of standard deviation, 13 vs. 24),
"70-300mm L appears to focus with the accurate group, while the 70-200mm f/2.8 IS is kind of on the border. Not in the teens certainly, but it’s the best of all the other lenses."
Except in sports, when you need dual cross types and cross types. And a wider aperture. General use, I'd probably pick the 70-300L because of the range. I'd never take the 70-300L to sports though.
If I recall correctly, that's also with the center point, not all AF points. I could be wrong though.
What is with the center point? The available dual cross types and cross type sensors available is with all AF points. In order to use dual cross typeS, you need one more than one point, right? Look at your manual, it's explained fully.
Read the LensRentals articles, it explicitly states that the center AF point was the only one use for the phase detect tests. My point is that while the article might show the 70-300L having more accurate focusing with the newer cameras, but it doesn't tell the whole store because it's only using a single point. So the AF accuracy might be quite a bit different between the 70-300L and 70-200L at different points, or using different groupings of points.
Now, that doesn't mean you are at all wrong about the 70-200L being a much better choice for sports in general, especially with the significantly wider aperture enabling the more accurate AF points and higher shutter speeds. I just wanted to point out that using the single center point doesn't specifically link to it being a better or worse lens for sports. It's the other factors aside from the more accurate focusing that makes the 70-200L the better lens.