The register adjustment screw is used to ensure proper timing between Bb and Thumb pad - both must close at the same time.
When only the "pinch" Bb is used, the thumb plate does not force the Bb vent to close, and thus the Bb vent opens and the register vent still stays closed as its spring is weaker than the Bb spring.
When the register key is operated and the thumb plate is closed as well, the Bb vent stays closed, and the rocker mechanism ensures that the register vent may open, provided the spring is OK and the pad doesn't stick.