Bingo. Your memory is dead-on. Take out that screw, and the shaft has a series of holes that let you slide and set the knob progressively closer to the plate.
Unfortunately for me, the screws are set as close as I can get them, and there’s still a lot of slack. It’s as if I need thicker decorative plates to fill the gap between it and the screw. But that seems like a janky solution.
