I have it set to basic Profile #2 for most things in the AF menu. I use the smallest AF points only, and only the cross type ones. I have it set so that it automatically changes AF points when you go from horizontal to vertical shooting. It helps a lot in fast situations. I also have it set so that the DOF preview button, when pressed and held down, changes the camera from ONE SHOT to AI SERVO, so it gives me the best of both worlds. Took me about 1 day of shooting to get used to that, now it just feels normal. And works quite well.
Any camera will have trouble, at f/4, this close to a subject, coming in and out of focus like that, towards the camera.... no camera system, other than manual focusing will really give you great consistent results. But the 5D Mark III does pretty darn good at it if you ask me, which is pretty impressive. I got off a few more shots of her that were not perfect or "good" focus, they were OK for focus, but it didn't take too many attempts to get it perfect. Also, her face or hair or whatever wasn't perfect for most, so this just worked well.
Good technique still is the best bet. I was tracking her with the DOF preview AI SERVO technique, and waiting for the split second where she's hanging in the air at the end of the swing motion, and then taking the photo. The 5D3 did very well. If I wanted to make it far easier I could have stopped down to f/6.3 or f/7.1 and taken a step back, and it would have been far easier. But I still wanted some bokeh background.