Isn't the pick up wired to the stator?

 

No.

The pickup is just a coil of wire wrapped around an internal magnet. One side of the coil is grounded to the case (and stator frame, and to the engine when all bolted up). The other wire simply passes through the stator frame and then out through the wire bundle with the other wires. It does not connect to any of the coils inside the flywheel in any way.