The Walton Tap Extractor was, as others assured me, a bust.
I did everything properly: I ground the tap flush with the surface, inserted the fingers all the way through, and made sure that the collar and inner shaft were kept tightly against the piece. Unfortunately, basic physics just doesn’t permit those tiny fingers to transfer the torque necessary to break the tap loose. I guess it’s just another $21 in tuition at Experience University.
Looks like screw adjusters it is!