Ceiling fan and lights not working after bulb pop
Usually a capacitor in a single phase motor circuit is to shift the phase of the start winding so the motor will start.

Otherwise a single phase motor will just sit there and hum since it will be locked up.

Here is a diagram of a ceiling fan circuit.

If you bypass or short circuit the capacitor I believe the fan motor will no longer start.

Better to replace it if it is broke.

If you disconnect it from the circuit and test it with an ohm meter a good capacitor will show zero at first and climb to infinity. Then swap your leads and it should do the same again. If it shows open or shorted it is broke.

Good luck and stay safe!