Thanks to a suggestion from one of the folks who left a comment below, K.G.Wuensch, we found the issue that led to the big discrepancy in the images you saw from my test, and the issue turned out to be with the D600, not the 6D.
Take a look at these images. Both cameras have apertures set to f/8, and I had one of our staffers hold down the Preview button on both bodies. As you can see, the defective D600 on the right doesn’t close the aperture down as much.
We found this behavior on two separate bodies. It affects not just the Sigma lens, but also the Nikon 50mm f/1.4G lens, so it’s not a 3rd-party lens issue. We know we received at least two duds, and the warehouse is looking to see if there are any more. We will, of course, take those – and any others we find – out of circulation and send them back to Nikon.