Ignition Issues - Car Won't Start Intermittently

Even easier test... Attach probe end of test lamp to brown wire at starter relay, clip end of test lamp to battery pos+. When operating the shifter the lamp should come steady on every time the shifter goes to park or neutral, since the neutral safety switch is simply supplying a ground path on that brown wire. If the lamp flickers at all in park or neutral positions, replace the switch.
There are a few plastic bushings in the shift linkage pivot points and those can fail and cause similar problems. I'll assume you've already looked at the overall condition of the linkage. Bottom line... no matter if the problem is mechanical or electrical, periodic adjustment should not be required.