Ideally, someone would recognize the symptoms and tell you what the problem is.
That someone isn't me.

In the meantime, I'd guess that the majority of bike light problems are related to the wiring and connectors. You imply that you've eliminated that possibility, however.

You could disconnect the tail light, just to ensure that it's not the source of the problem.

The headlight is the most complex item in the system, which suggests that it has the most ways to fail. The classic way to find out if it is bad is to swap in a known good one. That might be the hardest part of this technique. How well does your local bike shop like you? You could just buy a 6V, 0.5A bulb to wire in place of the headlight to just see when/if it lights up.

Another option is to run the headlight from a 6V, 0.5A power supply and see how it does. The complication here is that the light might need to sense the AC voltage to work correctly. An AC transformer with a 6V output might do the job, but the light might expect a current limited power source, so this has risks too.

That's the limit of my good and bad ideas. Maybe someone else has better info than I do.

Steve in Peoria