I doubt you have an alternator as such , you most likely have a magneto / rectifier with voltage regulator,
If you have 12 volts and no action from the battery it's going to be a safety holding you out or a bad connection
Jump across the two big terminals on your solenoid and if it responds by turning over , either you have a bad solenoid or a bad connection . The smaller terminal on your solenoid should read 12 volts when the starter switch is engaged , if not you may have a bad switch