Will a start capacitor be damaged by an overloaded motor?
The start cap being overloaded can be a symptom of the centrifugal switch not switching the start cap off when the motor is up to speed. Or the motor never getting up to the speed that would trigger the disconnect in the first place.

A start/run capacitor remains connected in the circuit (the "run" part of the name). So the motor being overloaded would mean more current going through the capacitor, which would then lead to overheating and overloading.