Re: MG MIdget 1967 - Starting Problems
Had a similar problem with a classic mini (same but different...) with no issue driving home until stopped for petrol then refused to start. voltmeter fitted showed good until trying to crank. Eventually proved to be battery-to-body earth strap had become poor even though done tight. Basically any point between battery, solenoid and starter, so worth giving them all a good going over.
Also not impossible for the starter to lose a winding/develop a shorted winding, although not had it happen myself.