Besides - magnetic pickups don't pick-up neck thickness.
Not directly no, but they do pick up the vibrations of the string and the sympathetic vibration of the neck and body and the mass of the components. There's a reason people put Bassass bridges on their basses!
Admittedly, I've yet to see a scientific study comparing a headless Steinberg compared to a regular big bodied Fender P confirming how much influence the mass of the instrument plays on tone. But my FatFinger definitely makes a subtle difference on my lightweight RBX4/A2