Richard, I hope you can benefit from the 2 years I spent trouble-shooting starting issues on my 782. When the key is turned to the momentary "Start" position, the battery terminal must be connected to the starter terminal and the ignition terminal --period.
If the switch is making the right connections OFF the tractor, it must also make the right connections ON the tractor.
So check to verify that the "Battery" terminal is connected to the battery, and that the "Starter" terminal is connected to the starter, and that the "Ignition" terminal is connected to the (+) side of the coil.
Finally, verify that the terminals at the "plug" are matching up the the terminals on the key switch.
This last step tripped me up for two years.
The copper terminals can be moved in the plug by using a small screw driver to depress the locking tab and pulling the wire out. This is sometimes done to allow a non-original key to work in an existing wire harness.
Good luck. I think that either you have an issue at the switch, or there is a problem from the key switch to the coil.
Edit: By-the-way, I think you're doing an excellent job of trouble-shooting the problem so far.