I removed the nut from the leaky flare, and look what I found..
Just as had happened with two other 1/4" flares, it snapped at the base. Maybe I turned the flaring tool too far, maybe I allowed the pipe to stray too far from centered in the nut while tightening. While cutting the pipe for a new flare, I noticed the flaring tool scored the pipe near the cut end. The edge of the cutter was scraping against the pipe.
I needed a straight section of pipe (no curves or bends) and needed to ensure the cutter was precisely perpendicular, not slightly offset on any small crooked part of the pipe. I wonder if this score resulted in a weak spot at the base of the flares..