Switch failures are rare, follow the following steps:
1) Inspect cable ends for wear, the brass barrels wear at the shifter & trans, carb & throttle, older boats have 4 of these, one on each end of each cable, replace if worn
2) Remove cable end from trans shift arm, apply a little grease on your finger & work it into the back side of the shift arm trying to force it back into the shift detent cavity.
3) By hand move the shift arm back & forth FNR, RNF, and back to N, a distinct detent will be felt, now test the starter switch, engine should crank, neutral safety switch is closed.
4) Adjust the barrel on the shift cable so it slides easily into the shift arm without causing the shift arm to move.
5) Shift cable is set, now test the throttle for correct adjustment, while someone slowly moves the throttle from neutral to forward closely observe the throttle cable at the carb & trans, (I like to put my hand on each) the trans arm should move forward fully into gear before the throttle cable begins pulling the carb. arm, if so your done, if not you need to adjust the throttle cable behind the throttle mechanism.