Make it as complicated as you like, but pretty much if you have the ability to hear the shutter cycling the wiring and power is likely fine. Change the bulbs and quit screwin around.
Bulbs should be the first check. If fuses failed or relays fail, fine, but not that common, AND for a fuse to fail it takes time to sort out.
Bulbs are known low life span items, just like front pulleys, fix it and be done. Not difficult, but if beyond your ability get help.
Thanks for the tip on cycling the hi/lo beans. I did that and yes, the shutter has power. Wish I'd done that first, but at least I've learned a few things about my bike I didn't know before.
Time to change the bulbs probably. What is the consensus these days on the best LEDs to get? I'm inclined to try Lamont's.