There is something wrong... most likely with your settings, but possibly with your camera. The camera should not default back to the center point (with expansion) when AF is started.
It should start focus with the pre-selected point at the center of the Expansion Points, wherever you have set that.
Here's the difference between Zone Focus and Expansion Point Focus.
Expansion Point Focus always starts at the selected center point within the Expansion, then allows the adjacent points to take over if the subject moves to one of them. But it should not always revert to the center point of the entire AF array.
Your camera has two Expansion Point Focus modes: 4-point and 8-point. The 4-point mode allows expansion to the first point right or left, above or below your selected "starting" point. 8-point allows expansion to all eight points surrounding your selected "starting" point, including the four on the diagonal from the starting point in addition to the same four that 4-point mode allows.
Zone Focus, on the other hand, is like a scaled down version of All Points, and focus can start with any of the active AF points within the selected Zone. The camera will usually first choose to focus on what's closest and covered by any of the AF points in the Zone (all other factors being equal).
Expansion is potentially more accurate, leaves less up to chance that the camera might focus on the wrong thing, but you do have to start with the center point.
Actually Single Point (no Expansion), manually selected, is still the most accurate. But when using it, it's down to you to put the AF point right where you want the camera to focus, and keep it there even when the subject is moving. So long as you do that, there's much less chance the camera will focus somewhere else.
If I were you, I'd try Single Point and see if the camera again defaults back to the center point AF point. Also try the other Expansion Point mode (8 points around the selected one, instead of 4 points).
It shouldn't matter whether you are using BBF or not, for any of the above. BBF does put you more in control of AF, but really shouldn't directly effect any of the above. But, just to be sure I'd also try your setup exactly the same except without BBF, to see if it acts the same way when the AF is started with half-press of the shutter release button. You also might try swapping the AF-On and * (AE-Lock) button functions, to see if that makes a difference. The camera should behave the same way in all cases. If not, something is wrong.
Next check to see if you have Orientation Linked AF enabled in your custom functions. I've found that if this particular CFn is on, but you haven't followed through and programmed the camera to use certain points and point-patterns in all three possible orientations, the AF system sometimes does strange stuff. Mostly it acts strangely when you change orientation... but I think I've noticed it cause other strange effects on my cameras, too (7D, not 5DIII... but it's similar).
Also check if you have set a "home" AF point (most likely the center one). Usually that's recalled by pressing the joystick (rather than tilting it) or using a recall button that's on some of the super tele lenses... But, who knows, maybe it's causing a glitch.
If you can't find anything in your settings, you might try resetting them to the defaults, or by fully rebooting the camera by pulling all the batteries (and pressing the shutter release button once to discharge any remaining power), or even by completely reinstalling the firmware. These things can "clear" fragments of misinformation that's gotten "stuck" in the camera's memory. If this sounds like dealing with computer problems, that's because it is. Today's DSLRs are as much computers, as they are cameras.
Hope this helps.