Ok, so, we assume that the battery is ok, and it looks like the charging system is working to keep the battery charged. I still would expect the idle voltage to be higher though. Maybe too big a diameter alternator pulley? Regardless, that wouldn't keep the starter from operating if the battery is good and charged. If you have a sudden no power in the start position, but, the lights and other accessories still work, I suspect the starter relay, the connections on it, or the ignition switch.
