As I recall from H.S. electrical shop class: The capacitor is used in many motors to compensate for a phase shift between voltage and current when starting the motor. There is a power factor due to the voltage and the current getting out of phase with each other. That means they are not in sync because the motor’s coils creates a current lag and the capacitor is added to create a compensating voltage lag.
Also since the motor starts and then stops, plus the AC unit appears to be old and could have many hours on it. Therefore I doubt the problem is with the capacitor. I would check the motor itself to see if the shaft or rotor, or bearing are worn out or it is vibrating which might cause heat build-up which might in turn activate the heat sensor which shout down the motor. You may need a new motor assembly.
I may be wrong, so let us know when you determine the cause of the problem. Good Luck