I wonder if it would pay to check the ignition pickup under the alternator cover.
They call it the crank position sensor, I think. You disconnect the plug in the loom with (for the Tenere, at least) the blue&yellow/green&white wires, and check the resistance. It's about 200 ohms.

We have trouble with these on our work bikes, and the symptoms range from those you describe to complete no-go.

They aren't a service item, and must be replaced with the stator unless you have a competent motorbike electrician who stocks these critters and can splice a new one into the system.