". The wire and magents make no difference to the saturation of the core at all."
Without the wire and magnets, you have no core period. It's all in the construction. Use a shitty weak magnet and thin conductors, you overload it easily. Use a strong magnet with thick conductors, you can't reach the saturation point as easily.
Pickup design 101. I've gone through several HUNDRED variations of magnet and conductor, from nail+bottlecap+bell wire+fridge magnets to United Nuclear magnets and 12-gauge wiring.
Pickups are not transformers, despite sharing several characteristics. Break out your o-scope and start building them.