Ceiling fan speed is not controlled by capacitors but by field windings. (The non-spinning part is the the stator. The 'spinning' part is the rotor or armature).
The speed selector switch merely adds more field windings to the stator windings thus increases the rotor speed. You can measure this on multimeter. Each speed setting will change slightly on a sensitive OHM scale as you change speeds. (Of course with everything unplugged and probes directly connected to the fan's input wires).
The 3-wire capacitor is actually two capacitors in one internally wired with both caps using a common lead. One cap acts as the 'starting' cap, the other as the 'running' cap. There ought to be two values printed on this double cap. (Possibly you can't see both values because they may have faded over the years due to heat). These are hi voltage (X2) capacitors and have to cater for the maximum voltage swing encountered in AC supplies (about 312V peak-to-peak for a 110V AC mains supply).
Therefore if your fan runs at only one speed, suspect the speed selector switch or most likely an open field winding(s). Do the multimeter test to determine.