Your example photo shows WAY too slow a shutter speed. Look at the little guy's foot! I photograph my kids playing at 1/640th if I want sharp images. Sorry, but his foot is not "planted". He's running!
That's not out of focus. There is a surprising amount of sharpness on his leading shoulder (considering the shutter speed). That is motion blur.
One thing to consider, at 70mm, f5 isn't a huge DoF for a moving child. Depending on your lens, you could have dropped to f4, or f2.8 and the increase in shutter speed would have yielded you sharper results. I'd bet your keeper rate would have been higher, even with a shallower DoF. And don't be afraid of ISO. That camera (I've had two) is pretty darn clean up to 3200. You just have to be more precise with your exposures.
With a narrower DoF (and suitable shutter speed) your focus accuracy will also be much easier to understand. The hair at the back of the head will be in focus when the eyes aren't, or the leading arm with be in focus not the face, or...
All that being said. It's a camera. You were in low, flat light (not a lot of contrast). Your camera needed to work HARD to track the kid! You need to help it any way you can. Brace your camera, move it smoothly, keep the AF point on one spot of your subject - a spot with as much contrast as is possible.
okay, all the technical (operator) stuff aside. As good as the 5D3 was I found that objects moving toward the camera were the hardest thing for the camera to track at close distances (like yours). I know people who can do it quite well with their 5D3's, but they have altered the case settings. I did not alter mine. For the type of shooting I did, I found Canons base settings to cover me well in most scenarios. But if you want to take photos with your kid running toward you, you will need to adjust the case settings to optimize.
Finally, the 5D3 was a seriously great camera for AF, but there are tricks and nuances that need to be worked with and around. For instance, that 5 point AF cross never worked for me. It seemed to lock on one of the outer points too easily. I always skipped it. I used either a single point or the 9 point square. The 9 point square (for me) tended to work best with subjects moving toward the camera as it seemed to prioritize whatever point was closest to the camera. Just make sure you keep the points on the face and not arms.
EDIT: I forgot. AI Servo all the way!