The neutral,start switch will be located on top of the control valve that you located by tracing the shift linkage to.
It will have two wires going to it,one white one yellow.....white one comes from ign. Switch.yellow goes to S post on starter.
When it acts up and wont start,check these things with a Meter......check voltage at the neutral switch at both the white wire then the yellow wire....then do a voltage drop test across the switch...check for corroded or loose connections at the ign switch at the neutral switch and at the starter.
The neutral switch has a ball on the bottom of it that rides on the fwd/rev shifting spool in the valve body,ball may be sticking or the spool may be wore some in the spot where the ball sits and rides,can unhook shift linkage and rotate spool 180 degree and rehook linkage.
Is a simple circuit so not much to it,is either corrosion in the connections or wires or a loose connection.