I just took a closer look at the picture of the brush assembly. Isn't that brushes, holder and regulator too?
If so, you may find the problem to be just a loose mounting/grounding screw on the assembly. IIRC the ground for the regulator and for the rotor, is one of the assembly mounting screws.
As such, if it's loose, you'll get no or at best intermittant output because the rotor can't energize properly.
At $10, the unit is about 10% of list cost, and about a 99% rebuild. Buy one just so you'll have it if needed, down the road. I have to assume you don't have a tach, if you're considering a rebuilt without a 'W' lead.
A lit battery LED when running, only indicates no output,because all they do is light on reverse flow (from battery TO alternator) which could be from any of the mentioned causes.
BTW, LEDs are generally rated at 100,000 hrs, so don't often fail. That's why they're used for indicators now, rather than bulbs, which, if you're lucky, have a 1,000hr life rating, and are usually a *** to replace. The circuitry is also simpler with the LED indicator.
I say again, DON'T OVER-TIGHTEN THE BELT or you will be doing a complete rebuild later.
AAA A lot of the brush holders also have a hole for insertion of a paper clip, or some such, to hold the brushes retracted for insertion. Then, you pull it out to release the brushes.
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