Most of your understanding of how pickups work is wrong, or at least incomplete. A pickup without a magnet sounds exactly like a pickup with a magnet. The pickup does not need a magnet at all, only a magnetized string! Now, of course, in fact, the easiest way to magnetize the strings is to integrate the magnet into the pickup, so that's why we do it.
Ironically, Seth Lover, the inventor of PAF, believes that these physics are absolutely correct. Seth Lover said in an interview with Seymour Duncan in 1978:
“…...the magnetic field comes up to the stings there and magnetizes the strings. That’s one of the things that most people don’t understand. They figure that string is waving there and cutting the magnetic lines of force. Nuts. That isn’t it. The magnet, all it does is magnetize the string. Now you’ve got a waving magnetic field. And we have a fixed coil with a waving magnetic field to induce voltage. If you want to, take the magnet out. One you’ve magnetized your strings, it will play until the string loses it. Players think the string, the magnetic field from the magnet comes up to the string and by twisting the magnetic flux back and forth that’s what induces the voltage. That's not what happens."
Therefore, we can see that the role of the pole shoe as a magnet and the role of the magnetic field generated by the pickup is greatly exaggerated. Our simple experiments show that the most important function of pole shoes as magnets is to magnetize the strings, and the most important magnetic field is the field related to the vibrating strings, not the static field related to the pickup itself. . In a sense, and for analytical purposes, as long as the strings are magnetized, we can effectively ignore that the pickup contains a complete magnet. Therefore, the most important role of the pole shoe in setting the tone is not as a generator of the magnetic field, but as a receiver of the magnetic flux radiated from the vibrating string. The importance of pole shoes as concentrators and filters for magnetic flux becomes more apparent in a chord-centric view. That's what I realized a few years ago, and that's what we manipulate in our pickup design to adjust the pitch. We design pole boots, using some familiar materials and novel and exotic metals that many others never thought of, to enter the pitch exactly where we want it.