My scooter isn't starting. My solenoid is about 1 yr old. (I replaced it 1 year ago.) . I can jump the solenoid (by connecting to the two nuts/bolts atop the solenoid with wire) and the motor will start.
The Battery gives a 12.5 Volt reading, but this doesn't mean the battery is not on it's way out. It only means the battery is fully charged. Even with a full Voltage reading, a battery on it's way out often do not have enough "holding power", to activate a starter motor.
In your case you actually need to do a proper load test on the battery first, to see if the battery is still any good.
Having said this, it also sounds like you might have a Voltage drop somewhere before the solenoid:
The scooter starts when you jump the solenoid - which means the solenoid is probably still in perfect condition. You could do a Voltage test on the thin wire at the solenoid. That wire gets power (+) feed feed via a longish route, thin wire, and switches when the starter button gets pressed. The problem could be located at any of those components.
It might be better to unplug the thin wire on the solenoid before you do the Voltage test at the far end of the loose wire / terminal. A low Voltage reading on that loose wire when you press the starter button should confirm whether you have a substantial voltage drop before the solenoid or not.
Should you find a substantial Voltage drop on the thin wire, it means the problem could be caused by loose / dirty battery terminals, faulty ignition switch, faulty brake switches, faulty starter button, or broken wires.
The starting point would be to do a proper load test on the battery first.
