The results you posted are in the range of acceptable. Unless you are seeing a problem in normal photography, I would not worry about the test results too much.
Zoom lenses - and the 24-70 in particular - have different AF Fine Tune results for different focal lengths. If you are concerned, I'd retest at 30mm, 45mm, and 60mm focal lengths or whatever focal lengths you use most often. It will give you a better understanding of what corrections might be required.
If you are trying to test for the sensor, you'll need to fine tune for that focal length even if your optimal fine tuning adjustment for all focal lengths is different. For example, you might have a small focus error at 45mm that is corrected with a -8 adjustment, but you could still decide to dial in a -2 adjustment for better overall performance at 30mm, 45mm and 60mm. There is an art to AF tuning.
Most of your test values were at the extreme focus points. The outer edges of the lenses are more likely to focus poorly with more AF errors and more likely to have lens issues. Unless you really use those outer sensors most of the time, I'd test with more of the central sensors. And in an evaluation of the lens, I'd use more of a weighted average approach based on the sensors you use and how often.
Check your test conditions. Lighting and WB can make a big difference in FoCal test results. If your AF Fine Tuning is a little off, the rest of the results will look far worse than if AF is optimized.
Don't be so quick to blame the camera. The lens could be a source of error just as easily as the camera - especially if you've had the lens for a while. If it really is a camera problem you will see a similar patter with other lenses and different focal lengths.
There are a lot of different tests in the FoCal setup. How did the camera and lens perform on the other tests?