Floppy / Brushless DC motor speed regulation
I wonder if the hall sensor is involved in speed regulation at all.


It's not. That hall sensor is only used to generate the 'index' pulse which indicates the start of the track. It produces one pulse per revolution.


  Could it be that it has MORE hall sensors below the flywheel


Yes. It will probably have 3 hall sensors between the coils. If any of these sensors weren't working the motor probably wouldn't run because the commutation sequence would be wrong. 

It also has an 'FG' (frequency generator) coil, which is a zig-zag track on the PCB under the outer edge of the rotor. If the FG track was broken the motor would speed up trying to match the reference speed setting, which is the symptom you have.

I suspect that either the FG track is corroded near the capacitor, or there is something wrong with your repair (broken track, bad joint, short) which is affecting the FG circuit.

Example FDD spindle motor with rotor removed:-

 

Example FDD controller speed control circuit (BA6486FS):-



  I cannot remove the flywheel to see where the motor is powered from,
  because the screwhead is b0rked but still firmly attached.


Try softening the glue with a soldering iron, then grab the outside of the screwhead with sidecutters and turn it. Note that this screw may have a left-hand thread!