It was operator error! Even though my voltage testers, all three of them, showed voltage when the breaker was off, when I put my multimeter on it and actually read the voltage, there was nothing there. So all of my voltage testers were reading the same thing, and it was wrong.
When I hooked my branch circuit wires to the breaker, the phenomenon stopped. All of the meters read correctly. It seems that when this breaker is unterminated, it has some kind of floating-ground type stray voltage associated with it that only the voltage testers pick up, and "no voltage" when the breaker is turned on! Weird! ������
This is a 100-amp breaker, and I saw the same thing happen with a 125 amp breaker in the same panel. However, this did not show up on the 15A and 20A breakers. When those were unterminated, all my testers read correctly - totally normal.
I seem to recall this happening once many years ago, where I was ready to send a breaker back to the manufacturer. But I forgot all about it - just another reminder to ALWAYS put a multimeter on stuff before freaking out! ������
Thank you all for your incredible expertise and help! ������