If the 2x4 trick works then it is a very sure indication of crummy contacts. If it doesn't work, really doesn't prove anything. A pretty good giveaway is that it sometimes starts, sometimes not, so it is not a 'hard' failure but a 'soft' failure. The switch contacts are about the only thing that would behave like this. A capacitor can fail either shorted or open and the vast majority of cases is a hard failure. Motor windings usually short or a wire breaks from vibrational fatigue. Another case of a 'hard' failure. Try cleaning the contacts.
Al