Yes, it seems that your 2006 alternator, Motorcraft MIGL665RM, was only used in '06, '07, and '08 3.0l engine, so a bit of an odd duck, not cheap either.
Rebuild might be a better plan as suggested.
Alternators have 3 "fields" that supply AC voltage to the diodes, diodes change voltage to DC.
Spinning rotor with voltage applied(electro-magnet) induces voltage in the fields built into the case of the alternator.
The rotor voltage is what decides the output voltage, the rotor has 2 brushes, the voltage regulator powers these brushes, it might send 7 volts to the brushes/rotor for a standard 13.6v output, or 9volts for a 14.6v output.
You are not getting voltage for free, i.e. 7v to 13v, the engine spinning the rotor supplies the extra 6volts
Generally brushes do get worn and can be replaced, in your case, with 12.5v that could be the case, low voltage at the rotor lowers output voltage, I would test the alternator output voltage after disconnecting the B+ cable on the alternator(the big wire), you are probably just reading battery voltage not alternator output, alternator output is probably 0v or well below 12v.
It could also be the voltage regulator itself, it is not sending rotor the correct voltage, usually there is a ground bypass on the alternator, you can put in a screwdriver and that will apply full power to the rotor, that would test if brushes and diodes are working.
The key switch provides the voltage that powers the voltage regulator, and that voltage powers the rotor, that's because the alternator would drain the battery if it had rotor voltage all the time, it is much like an electric motor so would suck power trying to turn itself.
The battery light or Amp meter is usually on that power wire that comes from the key switch, so if the light or meter is not working then the power might not be going to the voltage regulator in the alternator, so alternator is never "turned on".
Fields also fail, 1 field failing won't effect you much, head lights dimmer at idle, but battery would still be charged and maintained, and no.......head lights shouldn't dim at idle, lol, that is not "normal" that is either too small of an alternator or 1 failed field.