Have you considered the possibility that people shooting with a 1dx are just more adept (on average) at getting the most out of their gear? A priori, I think this is the most likely explanation for what the OP observes. A pro with a 1dx will take a better shot and post-process it better than a doofus with a 5d iii like me The reason you see more in the processed image is because there is more there to start with. I expect if you switched the cameras so that the pro had the 5d iii, you would find that the 5d iii magically started producing much better raw images (that could be pushed further).
Well, the OP has a 5DIII and a pair of 1D X bodies, and is therefore speaking from personal experience of shooting similar scenes (gymnasium sports with f/2.8 supertele primes, needing fast shutter speeds in the typically poor lighting of such venues) with both cameras personally. He's processing his own RAW files from the two cameras, and the needs of the shots often demand careful work in post (if you've shot basketball, volleyball, etc., in a gym, you're familiar with the need for high ISO, the crappy color of the inadequate-for-photography lighting, etc.). He's commented in several threads on the difference in how far he can push files from the two cameras.
So in this case, I don't think your explanation is the likely one.