The more I think about it, the more likely this capacitor is being utilized as a "Run" capacitor versus a "Start" capacitor. Start capacitors typically have a relay that disconnects the capacitor from the motor after start - a Run capacitor remains connected. If the capacitor wasn't up to spec or was failing this could have led to the motor not running properly (leading to overheating).
I'm standard AC, but mostly a DC guy when it comes to electrical projects. That said, my degree was based around electronics (Microprocessor/uController design and implementation - Industrial automation, etc). Which means I mostly know enough to get myself into trouble.