Seems odd. I could see the 24-120 failing to exploit fully the resolution of the D800 but not the 24-70. My 800 delivers stunningly sharp images with the 24-70 and, indeed, all my other Nikkors. I am somewhat at odds with the view you need to use higher shutter speeds with the D800. Why not the D7000 too, which has almost identical resolution? I have not found any need to adopt a stricter regime of shutter speed management with the D800. Without VR engaged (you don't have it anyway on the 24-70) the old adage of shutter speed numerically equal to focal length for handheld works just fine.
If your most recent tests show no improvement I suspect you need to have Nikon take a look at the unit. Years ago in film days I knew another pro who had a lot of Nikon, Canon and Pentax gear. When he bought something new - lens or body - he would shoot a roll and 9 times out of 10 send the purchase back with a very snarky note demanding they "finish building it" by re-evaluating the entire unit. He claimed quality assurance at the factory passes equipment according to a fairly permissive standard compared to what the equipment can actually be made to do. In the shop they can frequently improve on that with visible results even though the product passed QA. If he saw *anything* even slightly amiss, such as with shutter speed or exposure, back it went with that sarcastic note. On occasion he would return something on warranty even before taking it out of the box - right there in the store ("I don't have the time to test everything I buy."). He bought enough stuff I guess they would put up with this.
Bottom line: Marginally performing units do make it to the store shelf once in awhile.