If I recall correctly, I switched to AF-ON use when the D70 gave me a hard time in producing consistent focusing and found that it was easier for me to work by explicitly telling the camera when it is to focus and when not, independently of the shutter button. I also started using my other cameras in this way at the same time, although they didn't have the D70's focus problems. What I like about AF-ON is that if I want to do a spot meter reading of a specific area to set manual exposure, I don't want to camera to change the focus. When I put the camera on a tripod for a photograph of a static scene, I may set focus once and then I may do other things and I certainly don't want the camera to refocus every time I take a picture, it might do something wrong and mess up my shot. Yes, I can switch to manual focus but by assigning AF-ON to AF, and release+focus mode with AF-C, I can choose freely when to activate autofocus and when to focus manually without changing any switches.
Unfortunately things have gotten more complicated with all these focus-area modes and so that kind of destroys my principle that the camera should operate always the same way, without modes. These focus-area modes are sometimes needed for photographing moving subjects so I go along with that, even though it means sometimes the camera isn't in the right mode for the situation.
Accessing the focus point selector is easier when the thumb is free if you want to change the focus area during an active situation but I accept this as a compromise to get the other things that I like about AF-ON. I try to anticipate the situation and choose the right focus-area mode in advance so that I don't have to change the focus point or area during a situation. This doesn't always work but fortunately with the D6 I've managed to shoot hours of action without having to touch the focus-point selector (custom group-area with face-priority allows the camera to find the closest face within my selected area and this is typically where I have the face in focus in figure-skating, and it even switches automatically between two sets of settings between horizontal and vertical orientations). So only during one element (death spiral) did I have to move the group-area from the top of the frame to the bottom to get the shots, while for the rest of the time I could avoid touching any of the focus point or area settings.