Based on your diagram, it wouldn't start on speed 1 (you'd have to turn it to speed 2 to start, then back down). It looks to be what is known as a "bucking" transformer configuration (this article shows the general arrangement, in that case as used to slightly reduce the mains voltage) but while I've known of inductor-based controllers for (older) ceiling fans, this set-up is new to me.

The old cord definitely has to go but observing the level of rust, I'm not even sure if an acceptable earth connection can still be made. The capacitor value might be able to be worked out by trial-and-error (but watch out for dangerous series resonant effects).