The alternator supplies charging power to the battery. period.
It does NOT power anything in the car, the battery does that.
If the battery reading is above 12v with everything off, then check it with the engine running.
It should then read 13.5 to 14.5 volts, if not then the alternator is not doing it's job.
If the battery reads below 12v, but will take and hold a charge from a battery charger, then that again points to the alternator, if not then you have a bad battery.