Is the headlight charger useless if the green light is always on?

No. The headlight is red when charging. If it is green, it means it is fully charged or no power can be charged. It is best to discharge the headlight before charging it. This way you can judge whether the charger and headlight are good. If the headlight is not on and the green light is still on when you plug in the charger, it should be that the charger or the headlight is broken. If it is red, there is no problem.
The headlight has no power and the charger shows a green light when it is plugged in. This is caused by poor circuit contact. Check if the charger plug is broken. You can hold the wire and rotate it 360 degrees. If the red light appears once, it means the circuit is broken. If not, you need to open the battery compartment and check the two ends of the battery to see if the battery is displaced or rusted. Displacement means that the contact is not good. Just press it straight. If it is rusted, you need to clean the rust and then you can charge it normally.