Starting capacitor

The capacitor in a single phase induction motor is there to provide the direction of the motor by putting the start winding 90 degrees out of phase with the main winding.

After startup the main winding is responsible for the speed and power of the fan, in your case it sounds like the problem is with the main winding and it is running on the start winding alone.

Or it could be mechanical - such as bearing seizure.

I would not argue that a capacitor, might deteriorate slightly, but the basic capacitor philosophy is that it is a work/not work unit with any deterioration being undetectable, but that would not be responsible for this problem, if its working at all the fan will kick and turn, if it is not working the fan will hum and vibrate but will not turn.

There is no starting capacitor in a fluorescent tube light fitting unless you have a very old series ballast unit.

The capacitor in a fluorescent light is connected in parallel across the mains for power factor correction only, the light will work without the capacitor, but the crack at the switch will excessive when switching on.

Th small two pin round starter unit is in fact a thermal bi-metallic strip which incidentally does include a capacitor. This unit is a very basic type of timer switch, it is there to make and break the circuit to the ballast in order to provide the high voltage to strike the tube. Being mechanical, failure is usually due to contacts or the strip being weak. There is a more in depth explanation in this posted in this forum quite recently.

My ceiling fan nowadays, I have observed, does not spin that fast as it used to when I bought it. Is it due to the degraded value of the starting capacitor used, that it is not able to provide enough starting torque?

It could be mechanical. Check the level of detritus on the fan and in the bearing, and lubricate the bearing, perhaps? Does it wobble, indicating an imbalance?

Mostly the dry-up, [oil leaks or evaporate], Capacity falls & Fan slows & may not start,

If starting kick is low & fan stalls, a push to any side will start it in that direction

I would guess that the centrifugal switch is NOT taking the cap out of circuit after the motor is running, that is most likely the problem.

Such motors usually ONLY need the cap for starting, not running.