I will give you my take on this but after reading what Paul wrote and his personal opinion on this matter indeed there is little I can contribute to this thread. I will do the best I can though.

In the first place and, I did already discussed this with Nikon, why did you have to mess around with focus calibration? If you buy a lens under warranty and you see that in general your images are not in focus the proper step to take is to contact the manufacturer or general distributor of that particular lens. I have been using Nikon lenses for more than 50 years and I NEVER ever needed to calibrate anything. If the lens needs calibration I send it to the manufacturer. I have never used the Nikon 24-120 VR but from all I know it is a good lens and very capable of excellent images.

Most AF errors are operator's error. In addition and I have not examined your images to determine what you did but when depth of field is a necessity small apertures are in order. You do not have to lock the mirror for every single landscape you photograph but it helps a lot to make an image sharper. I also agree with the gentleman that has suggested that sophisticated software is not necessary to sharpen an image where proper technique was used. Cameras without the AA filter output sharp images and cameras with the filter are sporting filters which are pretty thin with just a minimal amount of imperceptible blur in their images.

Apart from the many useful suggestions you have already received I would like to offer another one. If you have a reliable photo technician nearby take your camera and lens to that repair station and ask to have your camera and lens calibrated or checked for calibration. This way you will make sure that if calibration is OK you will be at fault, not the camera or lens.

Finally, a tripod, even if using VR is in my humble opinion one of the most useful tools in photography when proper technique is used. Manufacturers of cameras recommend to keep VR off during the exposure but I have accidentally left it on without ill effects. A tripod and a locked mirror are the best assurance that an image will be sharp when we do our part.

If the OP isn't going to participate, I guess we can talk amongst ourselves. I'm confident the AF information has been stripped by the Affinity processing, similar to how LR will do the same. We'll get more about the first image with an unedited version. Answers to the several questions will further help fill-in context and background to how that image was created.

To your experience with BBF, whether the AF is active or not when the shutter is released is not directly relevant, certainly not for a static shot of these rocks. You can obtain focus and release your thumb from the assigned back-button focus. The camera / lens remains in focus as long as you don't change the position / distance of the camera to the subject. Alternatively, you can press and hold the BBF through releasing the shutter on the static rocks. Either approach should accomplish a sharply focused image straight out of camera. That SOOC in-focus image is what is missing from the first example and the issue and cause we should seek to resolve.

I have the exact outfit, (D750, 24-120 F4, 2017 Kit). Since new, I've had a Lenskins 77mm UV filter on it. At all focal lengths, and even hand-held to 1/20 sec., my photos with that combination have been very sharp and held up well to cropping. Occasionally, I have resorted to the MUp setting. It is always when the camera is on a tripod, and usually in a low-light situation, when timing is not important. The camera sits there with the mirror up for 30 seconds, then makes the exposure. Most of my shooting is on a center-point focus setting. If I was you, I might try every focus mode, with every focus point array and see if there is a clue as to which setting might be best for your specific subjects, before resorting to sending the whole works off to Nikon.

When I download and look at your first image in Capture NX-D, there are no active focus points. It's also shot at 1/20 sec at a 105mm focal length. This raises several questions:

1. How are you focusing? Why are there no active focus points?
1a. The image reports Affinity Photo, possibly this software has suppressed the AF information in the EXIF. We need an original version of the image, not using Affinity. Be sure to store the unprocessed JPEG attachment.
2. Was this image captured on a tripod or handheld?
3. You say AF-S. Does the camera beep a focus confirmation? If beep off, do you half-press the shutter for a second or so and allow the camera to focus?
4. Are you using selective focus with a single AF point or group / zone? For this first image, where did you want to focus?
5. Given the first image is not anywhere near in-focus, are you using the camera / lens Auto Focus feature?
6. Have you considered the ideas presented as: How to obtain sharp images in digital photography

You don't need software to sharpen out of focus images. Please provide details for the questions and an original image direct from the camera, without the EXIF being suppressed / stripped.