Hello, welcome to JustAnswer.
So you have power at the yellow wire at the key switch when you turn the key to start, but no power to yellow wire attached to the slave solenoid mounted on the motor, correct? There is no sound what so ever when you turn the key? no click?
Testing the neutral safety switch in the control is easy enough, Both of the wires that come out of the control box should have power when the key is turned to start and the shift lever is in neutral. The wires you are testing this way are going to be yellow. Not the blue, red, or green (these wires are for the trim, if the trim button is on the shift lever.)
Hard to get at wires.
Should the neutral safety switch interlock be closed when shift lever is in neutral fo start and open otherwise.
The switch on the back of the motor is not the neutral safety switch. That is the shift interrupt switch. When the shift lever is in neutral that switch should be in the open position. No continuity thru that switch when the motor is running.
But the switch you should be testing is in the control box that is behind the shift lever up at the helm. That is the one that has 12 volts on both wires when the key is in the start position. No the one back on the motor.
The control box is not the same as the control handle.
The part that is visible is the handle. The control box is on the other side of the side panel the the control handle is attached to. It will have the two cables that go towards the back of the boat sticking out of it.
The way that circuit works is all those wires equal a straight line from the key switch to the slave solenoid. If any of them are not connected, you may have just found the problem.
That yellow wire starts at the key switch, goes to the neutral safety switch then to the solenoid. That wire is also the one that makes the started work when you turn the key to start.