A whistling kettle would be cheaper, lighter, and easier to implement.
If you wanted gas to shut off the stove to save fuel weight I think it is hopeless task. How many grams would you save per boil? How many boils would need to save the weight of the device so someone would consider it worthwhile? If you want to shut off the fuel as a safety feature, then it might be doable… though still pretty complex.
I would suggest using the check valve on the fuel canister as your control valve. Your stove has a pin that comes down to open that up. Put a fitting between your stove and the canister with its own pin. That new pin has a mechanism that allows it to release. That mechanism would be connected to a thermal sensitive spring loaded device by a bicycle cable. The cable would allow you to route the thermal sensitive device up to your pot of water, where it would sit submerged in the water. When the water gets to the correct temperature, the spring pops, which in turn pulls (or pushes) the bicycle cable, which would then activate the mechanism in the fitting that then releases the push pin and the canister seals itself up.
Certainly just a conceptual design at this point.. I’m not sure anyone else can see the picture in my brain :)
What would the mechanism look like? I guess a sliding shim with a hole in it. In the “normal” position the hole is retracted. The pin pushes up against the shim and stops moving (allowing it to push open the check valve on the fuel canister). When the mechanism is activated, the shim slides over allowing the pin to pop up through the hole. The spring in the check valve pushes the pin up and the canister closes.