The purpose of AF-S mode is that the camera focuses until it decides it has acquired focus, then stops and you can do your recomposing which was the default way of doing things in the early years of AF when there was only a single point (or the peripheral points were not so sensitive). By default, AF-S mode is associated with focus priority. I guess when you combine AF-S mode with release priority, you should be able to interrupt the process of focusing by taking a shot before the camera has finished acquiring focus. I have never tried this combination of settings though.
I stopped using AF-S mode when I had the D70 and the resulting focus accuracy was poor. The camera would often decide it was in focus when it was not. In AF-C mode with AF-ON, I could wait until I could see the subject in focus and take a few shots, and the result was clearly better. In more advanced cameras AF-S worked much more reliably at that time, but I wanted to operate all my cameras in the same way, so I started using AF-ON with AF-C and release+focus priority (release priority if release+focus is not supported). This has worked well as I don't have to change settings if the subject is static vs. moving, and release+focus priority waits a little bit between shots to acquire focus before releasing, but does not interfere with focus + recompose if the subject is still, I can still use that to achieve more off-centered compositions. I monitor the sharpness in the viewfinder and take what I see into account in timing my shots. I'm aware that for static subjects AF-S with focus priority would probably give a higher percentage of focus keepers but I don't like the fact that the camera decides when focus has been achieved - I want to decide when the shot is taken, and I base that decision on what I see in the viewfinder. The timing may be slightly delayed by the camera if it wants to keep focusing a bit but this has not prevented me from timing shots based on focus and subject expression.
The bride (or bride and groom) walking on the aisle before and after the wedding ceremony can be one of the most difficult subjects to photograph. This is because the corridor has often poor lighting and behind the subject there can be bright light and structures (church altar, wall texture etc., or the outer door) which are attractive to the AF systems rather than dark faces in the center of the picture.
In particular the church I was at last time had brightly lit wall behind the altar and there was complicated texture in it (it is a medieval church). I think the safest place for the focus point in this circumstance would be to point it at the middle of the body (as then the wall carvings would not be near the active AF point, also center points are the most reliable in focusing), but I have a habit of placing the focus point on the face so that focus is accurate even when using fast lenses (f/1.4, f/2). In the corridor shots I typically use the 24-70 because it can accommodate different framings in a fast changing situation and focuses quickly. As usual my camera is in release+focus priority and I press AF-ON to keep focusing while I take the shots. Typically I get maybe one third of slight focus misses in this situation because there is not enough light and contrasty detail in the bride's face vs. bright textured wall in the background. Also the light level can be very low, ISO 6400 is not enough to get correctly exposed blur-free shots, and I've resorted to 12800. With the D810 this looks so-so, but with the D5 the results look ok even in colour.
Flash is another option for this situation. A flash unit typically has a built in focusing assist light but that only works (if I'm not mistaken), in AF-S mode with center point AF. I have read it being recommended that the aisle shots be made using the support of the focus assist light but of course then there is the problem that the bride (or bride and groom) may be moving more quickly than would be ideal for photography.
I am staying with the AF-C + AF-ON method but am considering moving my focus point to the center row in order to avoid distraction by the highlighted textures in the background. I suppose I could try the focus assist light method at some point. I just don't like the idea of having to switch to single servo focusing especially when the subject is moving. Incidentally the recommendation I've read for this situation also includes use of the flash itself to light the main subjects, but I tend to go with available light. It can be debated whether the flash fits in this situation (it would help reduce movement blur on the front side of the subjects, but a point light source adds sharp reflections and emphasizes skin pores which I don't like). I guess the available light can be regarded to be of poor quality as well, so it is a choice out of two not so great options. Anyway, I wanted to mention this because if the OP prefers to work in AF-S mode (shutter button activated AF), perhaps single servo focusing with focus assist light from a speedlight would work acceptably. I guess the key is how much depth of field you have, and how quickly the bride (or bride and groom) are walking.
