OK, so I got you to consider doing an electric test first. That was my only aim.
You are basically correct in your understanding of how it works. But why it doesn't work is important too.
First, you referred to it as "Selectric". That is Evinrude's version. Johnson's were "Hydro-Electric". Totally different remote controls, and either will work. It pains me to say it, but Johnson had the more reliable control.
When you put the control in neutral, 12V is sent to the neutral solenoid. and 12V is sent to both neutral and reverse for reverse. But suppose one of those 3 switch contacts is corroded or burnt and one of the solenoids is only seeing 3V in reverse? The solenoid won't hold the oil valve tightly closed and it may not be pushing the dog tightly against the reverse gear. See where I'm going with this?
Inside the gearcase, the #1 cause of failure is dirty oil. Not big chunks of rocks, but microscopic metal dust---the stuff that turns the oil black. The stuff that jams the piston in the oil pump. Once that happens, it is too late. Changing the oil won't help.