I would take the bulb out and check for continuity to make sure it’s bad. Other than that it sounds like you pretty much understand the whole function which I tend to think of as some sort of voodoo process - exciting the alternator/regulator to charge. Maybe need to drop a viagra in the
oil. But seriously you may need to take voltage checks to see if the alternator is idling at 12.8 volts or putting out the 14 or so necessary to charge the battery. There are some pretty good trouble shooting threads on the ERA forum on this including one for my car. Patrickt provided most of the guidance.