If the battery pack is at a fully charged Voltage level and the throttle's battery indicator shows that it is empty then the battery indicator may not be designed for the same Voltage of battery pack that it is connected to.
The charger port can be wired through the controller's charger port connector or directly to the battery pack. Either way will work well. The battery pack can typically only be recharged with a wall powered battery charger while the bike is parked and not being ridden. The exception to this is if there was a gas generator or solar panel on the bike that was wired to recharge the battery pack while the bike was on a ride. Some controllers have a regenerative braking feature which charges the battery pack during braking, however, the controller in your photo does not have that feature.