The neutral safety switch is designed to supply power to the start circuit in Park and Neutral. It is only to prevent starting in gear.
Your problem may be a slow or malfunctioning shift interlock actuator/solenoid. The actuator locks the shifter until a signal from the brake pedal releases it. When you turn the key enough to unlock the steering column, it doesn't allow the actuator to power up and lock the shifter.
One thing you can try is to turn the key on and step on the brake pedal a few times, listening for the actuator. It should give a click when the pedal is stepped on.
The actuator is located near the brake pedal switch. If you look up under the dash, you should be able to hear it when you push the brake pedal. The unit is kind of round with a linkage coming out of it.
On some models, there is an adjuster on the actuator that may become loose.
or it may be that moisture has gotten to it and spraying the linkage with something like WD-40 or some other penetrating oil may help free it up. Be careful not to spary the electricl parts of it, as you may do more harm than good.