Since the car was a race car, a pushbutton start is not surprising. Not a big deal.
To make the factory set up work, there needs to be power to the starter relay S terminal and a path to ground. Without opening the diagrams, S2 is probably start and S4 maybe the path to the neutral safety switch.
For the ignition to get power with key in start position, circut J3 is needed.
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In the bulkhead we see S2 and J3 but not S4. S4 could be for the seatbelt interlock in which case its irrelevant if you jumper at the underhood switch.
Next photo is the 'engine connector' and I don't see the J3
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looks like there's no J2 either
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