Hi,

Since the car fires right up and runs I would suspect a missing link in that circuit over to the ignition switch or to the solenoid.
It might be possible that you have a wire on a wrong terminal that carries power the correct way a round in the solenoid itself.
A solenoid wire(s) might be on there backwards. The terminals do look very similar.
I have seen that happen to me before.

Have you cleaned all of the fuses and their respective clips that holds them? The panel is on the drivers door hinge compartment.
Power that comes from the battery through a smaller wire to a terminal block has to be in good shape and be nice and clean too.
It takes about 50 amps to get the starter solenoid to pinch the battery for more power.
A locked up starter motor doesn’t change that fact as the solenoid is only a middle man for that main current draw.
So you are back to tracking what the wire from the switch is providing.

Phil