My DH guesses that charging systems on scooters are not a lot different from those on motorcycles.
The initial test involves using a good battery that is fully charged. Auto Parts Stores are good about load testing batteries for free. The age of a battery has no meaningful relationship to the quality. New batteries can fail and old batteries test good.
Using a multimeter on the 20 volt DC range, measure the voltage at the battery. A good battery will be ~12.5 volts
Place the scooter on the center stand and insure that the rear wheel is clear of the floor/ground.
Start the scooter and measure the voltage at the battery
It may fall to ~12 volts at idle RPM because the headlights may draw more power than the alternator can supply at that RPM.
Measure the voltage at various RPMs up to about 3,000 RPM.
The voltage should go up to about 13.5 but should never exceed 14.5 volts.
You may want to record these volatges in your manual if they are good. If bad, then you must check:
The grounding points for battery, engine and frame;
The stator (Alternator), the rectifiers (AC to DC)and the volatge regulator. For this, you will information that is in a service manual.
Just be sure that teh scooter doesn't go running off without you when you are reving the engine !