I've never tried the 5DIII, but I have a hard time seeing how you can be disappointed by the AF in the 6D. Granted, I don't do sports, and I always use the central point for focusing anyway. I used a 6D for a wedding after using a 60D for years. It's not just a step up, but a large step up in every aspect (so it should be a huge leap coming from a Rebel). The AF was always fast (enough) and accurate. The only time it was slow enough to be noticed was in very very dark situations, but even then the camera still acquired focus.
Here is a picture I took at a wedding reception (1/160 f/5.0). There is clearly motion, the ISO is somewhat high (1600) and the light is very dim. The couple is illuminated by my flash gun, look in the background to assess the ambient light and keep in mind that the flash did not assist during AF. The second image is a 50% detail from the area I was actually focusing on. The third and forth image (1/60 f/5.0 iso1600) is a situation where I could barely see the DJ with my eyes (all the lights are in front of him and all the light in the picture comes from my flash that did not assist AF), but the camera acquired focus anyway! In this case, I have to admit, it took a noticeable amount of time. If I had to give a ballpark number, I would say more than 1/4 sec.