Yes it could be just a bad cap. The centrifigual switch is what drops the start cap and winding out when the motor reaches speed.Can it be just a bad capacitor? I know single phase motor needed a push unlike 3 phase motor. Like capacitor with start winding? I think it's called a centrifugal switch or something like that.
Just ground out the terminals of the cap to discharge it. Then you can check it out with your meter to see if it's any goodI'm looking at $8 for a capacitor or $250 for a new motor. Never change a motor before and not sure if the capacitor is safe to touch. Don't want to get zap.