Why not use the feed wire that turns on the temperature light to trigger a relay to turn on the buzzer?

Because getting into the dash board instrument cluster is a pain in the rear on most any car.

 

I agree with marshallf3. A klixon on the radiator Normally Open is the easiest most direct way to go.

That way you need not disturb any stock instruments.

 

Another problem lives here....if the cooling system is short on fluid, as in below the level of the [ onboard ] sensor, it cannot measure the temp. of steam, so the engine will quietly burn up in short order. agree with marshallf3

Sons' '98 Escort did this when the waterpump started to dribble on a long trip. Warped head, and the whole other 9 yards. Repairable 'cuz the belt stayed on duty and the valves did not clash.

 

Upper area of the radiator will still get really hot even if the coolant is low due to convection through the metal. Of course we still haven't covered all the bases, what if the thermostat gets stuck shut?

I suppose you could find the full factory service manual for the vehicle in question, locate the sensor on the engine that operates the warning light and connect to that, however some modern vehicles run all this stuff through the engine control module so it may not be a simple switch but rather a variable resistance sensor in which case you'd have to hack into the wiring somewhere.

 

On most cars with a ECM the coolant temp sensor for the ECM is different from the dash warning light. By that I mean there are two separate sensors.

By running the warning buzzer relay in series with the dash light you wouldn't have to get into the dash. The circuit goes; +12V - light - (add relay coil) - sensor - ground. The sensors are single terminal devices.