I had a 70-300 f4-5.6 IS USM for many years before getting my 70-200 f2.8L IS MkII. When I got the MkII, I took many comparison shots and, without a doubt, the 70-200 is sharper. One thing I wanted to know is whether I should continue to carry the 70-300 for the extra reach. But, cropping shots taken with the 70-200 at 200mm is still sharper than the 70-300 at 300mm.
I have also shot figure skating events with the MkII on a 60D and learned that the 60D is a bit lacking in the focus tracking department. Roughly 20% of my shots were OOF. With the 7D, less than 5% are OOF.
First tip for shooting sports with the 60D/70-200 MkII, make sure you're shooting with center focus point only. This will prevent another subject from grabbing focus from a different focus point. Also, the center focus point on the 60D can take advantage of the 2.8 (even if you're shooting at smaller apertures).
Second, use AI Servo mode. Be aware that AI Servo mode is predictive focusing and not exactly real time. It tries to predict where the subject will be when the shutter is actually tripped. To do so, it keeps track of the movement of the subject.
This works great when the subject moves in a straight line, but if the subject takes a sudden turn, the predictive focusing can be thrown off. This is where I ran into trouble with skating. That first shot or two just after the skater turned were more likely to be OOF. One work around is to get in the habit of momentarily lifting your finger off the shutter button when you subject changes direction. This forces the AI Servo mode to start over.
Third, look into "back button focusing" (google it). This is one of those things that you'll either love or hate.
Fourth, make sure your lens is in IS mode 2 for panning. Then, keep your elbows in, turn at the waist, and continue panning through the shots. (I don't agree that IS doesn't matter at high shutter speed action shots. Faster speeds may mitigate some of the benefits of IS, but I think every thing you can do to eliminate camera movement helps.)
Fifth, don't be afraid of ISO 3200 for action (noise Ninja helps a bunch).
When I first used the 70-200 on the 60D for skating, it did seem that I lost more shots to OOF. But, it then occurred to me that OOF was more obvious at f2.8 than using the 70-300 at 5.6. One solution was to shoot the 70-200 at 5.6, but I really needed the 2.8 for low light rinks.
The 70-200 f2.8L II is a great lens, but it may not live up to its full potential as a sport lens on a 60D. It's a fantastic sport lens on a 7D.