… Once it is up to speed it works just fine. …
That would indicate it is not a run capacitor.
That it needs help to start rather indicates a start capacitor, winding or centripetal switch problem.
You should hear a definite click as the motor runs down (close towards stopping). That generally indicates the switch is working (but not always).
Continuity through the switch can be readily proved by a quick resistance check. Open circuit would indicate switch contact problem or a break in the start winding.
The usual problem is the start capacitor has failed completely or has gone low on capacitance. A common issue.