That is correct, simply connecting AC across the winding would not make the motor turn as all it would try to do is assume one of two positions as the polarity changed. There are actually several windings alternately positioned around the rotor, with all the even numbered ones in parallel and odd numbered ones in parallel (or each in series instead of parallel) . One set is fed directly from the AC, the other through the capacitor. The phase shift provided by the capacitor (if it's the right value!) makes the current peaks in the windings occur at different times so the rotor is pulled from one to the next with the time delay causing rotation.
Obviously if the capacitor value is wrong, the time delay is too long or too short and the pull from one position to the next becomes unbalanced.
Brian.