My theory, FWIW... was that the disconnect relay itself does not move unless the switch is operated, all contacts are in their normal released position. When you plug in, it doesn't move, everything works according to the OP or when you start the Genny, it doesn't move and again everything works, Unplug, and again, it still hasn't moved but 12V battery power distribution fails.
In not moving, 12V power from (the Batteries) should travel over those same normal contacts but does not (ie it probably never reaches the relay contacts) where as in the other two modes, both furnished from 1 source, the converter, it does.... Question is.... why?? The answer is the direct path from the batteries is open somewhere whereas the path from the other source (the converter itself) is not... (I cannot remember which older model converter uses this concept, perhaps one of the Magneteks...)
The distribution of 12V battery power is over a released relay in the converter in some units. When plugged in, that relay operates allowing the 12V generated by the converter to reach the distribution network rather than the actual battery 12V. The battery is then charged by a second set of wires and operated relay contacts in this case. If those relay contacts should fail (ie be stuck/welded in the operated position) while in the relay itself is in non operated position, Guess what???