For the record, I solved the problem. The problem turned out to be a combination of things. The issue was the headlight was not working when I turned the key to lights. First of all, the motor DOES need to be running to power the headlight.
Looking at the wiring diagram, there are three things that get power from the lighting coil via the GREY wire:
1. Headlight
2. Tail light
3. Speedo back-light lamp (not the neutral lamp)
First thing to do is check all the bulbs for broken filaments.
If none of the above lights work while the engine is on and the key is turned to the third position, the problem could be with the lighting coil or the main switch itself. Check the three-way connection at the main switch GREY wire and the YELLOW wire from the lighting coil to the main switch. My first problem was the yellow wire had come un-soldered.
If the speedo backlight works, check that the ground wire on the headlight unit is secure.
It turns out that my biggest problem was bulb seating. My model BS-7s has a fairly large 6v bulb with two contacts (one for high and low beams) and a ground through the case. The bulb is attached to the bulb holder by three flattened metal pegs and a spring compression. The problem with mine was that the spring was so strong that it was hard to turn the bulb enough to seat the bulb without breaking the glass. You have to make sure that the bulb is turned as far as it can go so the contacts line up. If you have the model you know what it looks like. ALSO, the bulb is directional. It should be marked on the metal part which side is UP. Once you install the bulb you can twist the whole unit before you clamp it down to the headlight housing.
What I had to do in the end was take a dremel and a cutoff wheel and grind down the pegs underneath the heads so that the shafts were narrower but were still able to hold the bulb securely. I can take pictures if anyone is confused by that.