Oh yeah -- I nearly fell down yesterday... but on the
unclipped side.

Look for the *key point*.
I had stopped, with my left foot unclipped as usual, and was standing there with my left down and my right still clipped in, bike leaning to the left. I can't exactly remember what I was thinking, whether I wanted to move my bike or was turning around to look at something. But, I was shuffling my left foot closer under me -- imagine straddling your bike with both feet down, putting your left hand on the saddle, and twisting to look to the rear while your feet shuffle around.
I put my left foot far enough underneath me that it was almost past my center of gravity. *If I was using plain pedals, I would have been able to easily stand on my right foot.* But, putting weight on my right would have flung the bike to the right and surely thrown me down; if I tried lifting my left foot off the ground, I would lose the only support I had. I gracelessly hop-shuffled my left foot away from the bike, thankfully avoiding a fall.
As dumb as I felt about my first 0-mph fall, I would have felt even stupider with this one.