From your description - sounds like a short in the wiring in either the ignition switch, integration relay (sort of the car's central tie in point), or the starter relay and associated wiring. Depending on where the original damage was located, could be the insulation around the wire (wire jacket) has worn through and making contact with the chassis or wire could be crushed somewhere - either one would act intermittently, which would make it that much harder to track down.
Fortunately, it sounds like it is just engaging the starter - hence the starter cranking without the cylinders loading up with gas and actually firing up and running. That will help limit your search to the starter relay (in the relay box by the airbox, on the driver's side fender) and the wires that are hot for ignition switch position "start". If those check out, then you can expand the searching into the integration relay block and due a quick visual inspection of damaged wiring.
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