Ok, this trick worked as advertised, almost magically!!! Was sceptical at first, but seeing all those replies convinced me to at least try! I had to blow the air just at the entry of the headphone jack, pointing downward, and that's when I heard the click.
Thanks a bunch!
However, I had the same H/P Off message with the click, so obviously, seems to me that it just detected that a headphone had been plugged in the whole time, shutting down the volume of the speaker. I'm almost certain that plugging and removing real headphones may have the same effect, too, just by resetting the H/P state. That would be worth a shot next time... If that doesn't work, it means the dust bunnies are just holding on too tight!