I am going to dive in here feet first and give you something to look at.
The two brown/yellow wires at the regulator WL are linked either crimped together or linked at the Lucar connectors on the regulator.
One wire is direct to the gen warning light the other goes to the cintilla thermal delay switch which energises the choke solenoid to keep the carb induction flap closed on initial cranking. The choke solenoid when energised on initial crank will not actually pick up and hold the flap unless you go through the procedure of pressing the accelerator pedal down before cranking. Choke will stll work as that comes off the manual choke coil which is heat related from the exhaust manifold.
That aside.... it is quite possible you have a fault inside the cintilla thermal switch as these are a problem......I have lots of faulty ones but have not found a suitable repair method yet. See the picture below.....Jean7 if you are reading this.... one day I will
. I have been able to repair some when it has just been oxydised contacts and the elements and insulation good.
My thoughts here are maybe the element wire has broken (which they do), and making contact with the bi-metal blade or the insulation on the blades has broken down (which it does). As there is an earth circuit on the thermal blades if the element has broken away or insulation rotted through you may be getting a circuit through the gen light that is keeping it alight.
Just something for bedtime reading and following the schematic before you start at the weekend to get your head around.
What I would do is (ignition OFF) and the brown/yellow wires removed is test the wire and see if it does go to earth. If the brown/yellow wires are separate then you can test each one individually disconnected from the regulator. If the light goes out with the choke thermal switch disconnected then for sure this is now where to start looking. Unfortunately the thermal switch is tucked under the metal housing with a lot of other relays. I can see you have speed control which makes it even harder to get to.
Do the quick wiring check and see what the results are and let me know.