BJ8Healeys
Jedi Warrior

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By "throttle switch" did you mean throttle position relay or dashboard switch? I think you probably need a dashboard switch but your throttle position relay may be working. You can test throttle position relay by applying 12V and ground to the relay (normally this happens when you turn on ignition but your wiring may be changed). Make sure the linkage is connected so that the lever goes up and down with the accelerator pedal movement. Check the output as you press the accelerator pedal down. The output should show 12V at about 30% pedal travel. IMO, bypassing the throttle position relay would be fine since Triumphs were shipped without one.
The "throttle position relay" is not a relay, but a simple switch. It has two contacts inside that are opened and closed by a D-shaped cam that rotates with the arm that is connected via linkage to the accelerator pedal. The only purpose of the throttle switch is to keep power on the overdrive relay coil (which keeps power on the overdrive solenoid and the overdrive engaged) when the dash switch is turned off. The overdrive then only disengages when the accelerator pedal is pressed down sufficiently (which rotates the cam and opens the switch contacts) to ensure that the engine is driving the overdrive, and the overdrive is not being driven by the rear wheels. The throttle switch can be disabled, but then the "reverse torque" protection to the overdrive is also disabled. Shifting out of overdrive then via the dash switch can be very harsh.
I installed a dash warning light in my BJ8 to indicate when the dash switch was in O/DRIVE position (not when overdrive was actually engaged). I did this soon after buying my BJ8 30 years ago when I kept reading warnings about not reversing the car in overdrive, and I kept finding myself doing it. The light reminds me to turn off the overdrive before backing up, which came in handy once when my overdrive refused to disengage. I suspect that Danne's light was installed for the same reason, although his actually indicates if the O/D solenoid is operating or not. My light has been a useful troubleshooting tool when the overdrive has failed to engage on a few occasions. It has allowed me to determine quickly whether the lack of overdrive engagement was caused by an electrical problem upstream of the dash switch instead of a hydraulic problem.
The overdrive solenoid pull-in coil is about 0.5 ohm, which should draw about 24 amps for a short time, then the holding coil is about 12 ohms, which draws 1 amp. If the solenoid internal changeover switch fails to function, then the pull-in coil will continue to draw 24 amps until the solenoid or wiring burns out. I had that happen once and the solenoid got so hot it fried its rubber boot. I was fortunate enough to find a Jaguar solenoid (minus the plunger) on eBay for ten bucks. It's still working fine, but I put a 20A fuse in the overdrive circuit just in case.