Based on this info, it sounds like the switch as failed. At this point with it not starting the issue is most likely the PCM doesn't know the shifter is in part so it won't allow it to start. The PCM has to see Park or Neutral in order to start the engine.
If you have a scan tool that can monitor live data then look in the PCM at ignition status and see if it says park or neutral when in park or neutral. If it doesn't then the switch is the issue which means the shifter needs replaced.
The only other issue could be the range sensor but since you said you were pressing the interlock solenoid on the shifter to release the shifter that makes me think the shifter is the issue. You said you were pressing the neutral safety switch but that is called the Transmission Range sensor on this vehicle and that is on the transmission itself. The solenoid on the shifter is the shift interlock. That locks the shifter in Park unless the brake is pressed.
Let me know if this info make sense and we can go from there. Please correct me if I am not understand something properly.
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Saturday, April 11th, 2020 AT 8:45 AM