AF-ON with AFS

What you describe at the beginning of your post is a situation where the camera should be set to release priority - then you would not miss the shot.

Why would you set the camera to focus priority if you do not want to take the shot where the camera is focused?

To me the whole point of having a choice between focus and release priority is that the camera will behave differrently according to your chosen priority. That was how earlier models (and some current like the D7000) worked, and before the D4/D800 I never heard any complaints about it - because it worked the way it was supposed to.