Also if you can, remove the box from the side of the console & lay it flat before you undo the screws - stops bits of mechanism dropping out when you take the lid off!
Not that I have any experience of this scenario ....
Jumping back to the battery switch corrosion, the issue is only seen on high current draw. I'll bore you with ohms law of things in series if you want, but suffice to say the resistive contacts don't nick as much of the overall available voltage when the current is low. E.G. you might hear the choke solenoid go "click" but then try to crank & nothing happens as the starter pulls the best part of 40A. Remember a PT motor can drink a fair amount of juice as well...... Poking it with a multimeter will be at almost zero current.
New neutral switch can't be a bad thing either way.