But truth be told, 6FPS is nowhere near good enough for sports either, so 6D or 5DIII - FPS wise only, are in the same boat.
As I mentioned in the first post, I have learned to catch the right moment in volleyball and football due to lack of FPS on my current 550D, so leaning more towards ISO and image quality performance.
People that tells you there is a noticeable difference in regards to IQ is delusional, as 99% of IQ comes from the lens, not the body (well at least between the 2 in discussion).
ISO difference is again negligible. 1/3 of a stop, big deal...
I take better AF (which is worlds apart btw) over same IQ & ISO difference
Sure 5D3 isn't a full-fledged sports camera like a 1D4 or 1DX, but it's a hell lot better for the task compared to the 6D.
And good job of you ceasing the moment.