Sorry to disagree with the diagnosis.
If the Neutral safety switch is bad, there won't be any click noise.
The click and no start is more likely a bad contacts at the starter solenoid.
If the starter positive terminal where the heavy gauge wire attaches can be reached (the terminal which goes to the starter motor); a jumper to that terminal with a heavy duty wire will confirm the bad solenoid.
If you have a Toyota, there are repair kits to overhaul the solenoid.
Replacing the neutral switch is not an easy job, depending on the vehicle.
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