Dennis,
Yes, irrespective of switch position, when the inverter/charger sees 120 VAC from either shore power or generator, it automatically does two things. "Passes through" the 120 VAC to all circuits downstream of the inverter/charger AND turns on the battery charger section. The switch only functions when no external source of 120 VAC. In the off position, the inverter is not in the "standby" mode which does take a small amount of power. In the on position, the inverter is in standby mode and ready to power circuits downstream of it.
In troubleshooting, I would start by turning off and back on the 120 VAC breaker in the main breaker box that feeds the inverter/charger. If that does not do it, would check the breaker(s) on the inverter/charger itself.