Check and clean battery cables, then test battery under load as well as alternator output. Once you’re sure that those are in proper operating condition, if the engine will not turn over, then it could be the VATS. One thing that stood out in your post was that the car shut off while it was running. A VATS related problem should not cause that. Most VATS related problems are encountered on initial start. If the car started and was running, then the VATS system test passed the security check point.
Two weeks ago, I had a 94 LeSabre with the same problem, however, the owner (relative) drove the car home from a routine trip and all was fine. The next morning it would not start (turn over) no clicking, or anything. The lights would work until you tried to start it, and then everything would go off, except the security light. I checked the battery voltage and had 12.4 V. I then checked the cable connections for tightness and they were tight. Tried it again, and nothing! I then removed the negative cable and it was clean. Next, I removed the positive cables (two piggyback) and they had a small amount of corrosion where it contacts the battery. I thought that is not bad enough to cause this problem. I then pulled the two cables apart and found a small amount of corrosion at that point, so I got the soda water and a brass wire brush and cleaned all the parts of the terminals and reinstalled them to the battery. It fired up the first try and has not given a problem since.
The point of this story is to emphasize the importance of clean connections. It does not take much corrosion on the side mount terminals to cause a problem.
I remember back in the days when all batteries had top post and with an inch of corrosion around the terminals, they would still start the car. We would just pour some soda water on the crud and wash it off!
