I'm also leaning towards the fact that your camera is busted, and needs some TLC at service center.
Some reason it seems there's nothing in focus in your pictures.
Could you try one test, use the 70-200/2.8 at 2.8, and go low on the grass. Angle camera almost horizontal, slightly down. Grass should fill most of the view, maybe some sky on top. Focus some 2m/6ft distance. Is there certain spot on the grass that looks sharp? Manual focus is ok for this, just dial ~2m/6ft and you're good.
If that doesn't get any grass to be focus, then it's not focusing issue, but something is really wrong.
One really far fetched guess, maybe at DPP there's some setting it'll apply for 6D photos, unknowingly for you. If you share one of the live-view shot RAW-files, me (or someone else here) could check it and make sure it's not DPP. I'm 99% sure it's not that, but there's always that 1%.