Are you sure you are looking at the right card? This seems like a simple user error like looking at a CF card, but all the images are on the SD card.
If you formatted the card, you should not see any images. This is what leads me to think a simpler user error.
If its file system is doing its job without errors there's zero reason to reformat it between shoots.
formatting isn't going to hurt anything and they're zero reasons why he shouldn't format between shoots.
The 5D3 format menu says "low level" format, but all it really does is a reinitialization of the card (zero-fill).
Each time you format that card you may just be doing a high-level format, which means errors can slip by instead being detected and either corrected or marked as bad sectors which can no longer be written to.
And just deleting the images doesn't provide anything as well.
Not to argue terms, but low level formatting is not what you think it is. Low level formatting is done at the factory and not even possible on modern hard drives without special factory utilities. Reinitialization is what zero-fills the hard drives, and what people confuse with a low level format. A "quick" format only removes the pointers in the FAT (file allocation table) and is typically called high level formatting. Disk checks like checking for bad sectors can be done in a high level format or reinitialization format.
back to the OP..
Are you sure your 5D3 is writing to the card you think it's writing to? When you pull a CF card and leave a SD card in a slot, it will default to the SD card for writes and playback. You have to switch it back to the CF card.
I would suggest formatting the card in camera, make sure it is writing to the correct card, and shoot again.