Here's the last bit of info I think I will squeeze out of this experience:
Finally came across the old CDI unit in one of the boxes from my move. Went ahead and installed it in place of the new unit to see if it would work. It started buzzing when I turned on the ignition, and bike would not start. Reinstalled the new unit, no buzzing, and it fired right up. So... if the CDI unit itself is buzzing, then it may be on its last legs.