Finally got a chance to look into this, and fixed it.
Turns out you can remove the linkage that connects the two enrichening circuit plungers without removing the carburetors. The actual process involves a bit of sleight of hand and requires you do things in exact order. It reminds me of my wife's ability to remove her bra without taking off her shirt- not quite certain of the sequence of steps involved, but the results are both surprising and pleasant.
At any rate, once the linkage was removed (don't lose the nylon washers or the spring-loaded detent ball) the enrichening plunger can be removed with a 12mm socket. I blasted out the passage with some carb cleaner using that flexible red straw that inevitably seems to disappear after the first use. Then I just screwed the enrichening plunger back into the carb body and replaced the linkage by reversing the removal process (involved the removal of the left carb's circlip and spring on the plunger and some fiddling best left to those of us with three hands, but eventually everything fell into place).
The end result? Bike fires right up on cold mornings, no drama, no endless restarts.
Okay, the wife says I've spent enough time web surfing this evening. She's performing some sort of contortions with her arms inside her blouse.