It seems unintuitive to me why putting a stationary hunk or iron or neodynium in the middle of a copper coil generator would increase its conductivity, because the metal could act as its own conducting path and short paths along the copper wires and furthermore cancel out whatever opposite magnetic field is moving around it. So, why does it help?
The magnetic material doesn't increase the conductivity of the stator's coil.
The principle of any electrical machine is to use a rotating magnetic field to generate electrical coupling which allow for energy conversion. The stator's current is just a reaction to try counte-attack the induction in the stator's coil(lenz's law).
The reason why these materials are call "magnetic material" is that either they can generate a very large B field for relative small current or that their electron's spin are naturally aligned creating a noticeable B field.