If you have short listed 5D MK III and the 70-200 L IS II, I see no reason why you should be concerned. Like thomasl4 (the person who originally made that video) said, that "the video isn't meant to scare off potential 5D III buyers, because in fact I love the camera. It's been one of the most exciting bodies to use for Wedding Photography" ... just like your camera was stolen out of bad luck, thomasl4 too got one of those very rare quirky 5D MK III's that sneaked out of Canon's production line with a minor defect (if I may call it that). After watching that video I tested my 5D MK III with the 70-200 L IS II and I did not find any such problems, it was rock steady. Don't be too concerned about that video (it is one of those very rare problems), your choice to go with 5D MK III & 70-200 combo is excellent ... you should go fit it.
Sorry to hear about your stolen camera ... I recently lost my brand new 24-70 f/2.8 MKII, so I can somewhat understand how must be feeling, but mine was only the lens, but you seem to have lost the images too (don't you transfer your images to your computer everyday? ... especially on vacation I transfer all my images to my laptop at the end of the day, as well as make a back up in a portable hard disk).