When you are cranking over the motor, there is an output on the starter solenoid that sends full power to the ignition system and then when the key is put into the "RUN" position, this power out of the starter solenoid is turned off and then the ignition system is powered off of the ignition switch on the column.
Since this ignition output on the ignition switch also powers up other things on the car, by doing some more observation of things one can narrow it down as to whether the problem is with the ignition switch or possibly a connection issue through something like the bulkhead connector.
Since the "RUN" wire off of the ignition switch also powers other things if one were to turn the ignition switch to the "RUN" position and the TEMP and OIL were to come on then I would think the ignition switch is working so then I would be checking for power out of the bulkhead connector that goes to the ignition system.
On this below diagram, there is a PINK wire off of the ignition switch that eventually goes to and through the bulkhead connector and out to the ignition coil but if you look at the diagram I failed to trace out how it branches out to the fuse block to then power other things up. Also notice too on the diagram it calls out a carb idle solenoid that your car may or may not have and it also gets power from the same source off of the bulkhead connector.
And here is just the circuit going from the ignition switch out to the coil and then also the coil connecting to an output on the starter solenoid: