This D600 forum post might help: https://www.flickr.com/groups/nikond600lowlightgroup/discuss/72157631964731927/
I shoot with a D800E and the Nikon 24-7omm f/2.8G. I do need to set up for backfocusing issues, maybe a -11 set into it now. Nikon has worked on it but seems it is "Within factory specs and I can adjust it out." If I fire off a battery of shots while on a tripod at a fixed object it always seems one shot out of 5 will be far sharper than the others when I get into a 300-400% magnification in post. Press the shutter button each time and I can see the lens spin off a little bit like some sort of plus or minus tolerance level. Seems the nature of the beast and even color will throw it off, like using the red focus assist lamp in the Nikon flash over using brighter daylight, etc. Sometimes the first shot is the worse of the lot, even turning it on and first shot of the day is a clean miss. Guess the lens focuses a bit better on the second try (or stiff?) as it gets closer (maybe?), but not with 100% guarantee either. When it hits, it's pretty amazing.
You might try that Focal software too. It puts out a chart and you can see all the shutter trips and how far they wander from a focus sharpness curve. If it takes a bunch of shots at the same setting (like 5 or 6 shots against 3), it is telling you that it is having a hard time determining a sharp shot out all those shots and something isn't settling into a sweet AF spot. I've never seen a smooth line with all the test shots stacked on top of each other. My 105mm f/2.8G is really bad for AF and seems that Focal software takes a whole bunch of shots with that lens trying to come up with a best sharpness curve.
Aside, I use single point focusing most of the time too.
Your issue sounds quite a lot like what I'm getting with my D600. I just did a bunch of tests on both bodies and one is way out but the "good" one is pretty much on but yes still inconsistent. I wonder if a sencond hand D3x is the way to go?