I have to ask a stupid question after having posted my last message. This IS an electrical gauge isn't it? You are not talking about a capillary tube mechanical temperature gauge are you?
Assuming this is an electric gauge, if you saw no needle movement when you shorted the green/blue wire to ground, the problem is not the sending unit. Shorting that wire to ground allows the maximum current to flow through the gauge and simulates what a really, really, hot sending unit would see/do. No movement on the gauge when you bypass the sending unit means the gauge, voltage stabilizer, or wiring is bad.
After failing the sending unit bypass test, the gauge should not have moved at all in your boiling water test. To test the sender in boiling water it is necessary to leave the green/blue wire connected to the sender AND connect a ground wire between the sending unit body and the car's chassis ground. If you do not provide a ground wire between the sending unit body and the car's chassis, there is path to ground for the current (nothing to make the gauge work).
Again assuming this is an electrical gauge there are a couple of other tests you can do but they require more work. Remove the temperature gauge from the dash but leave the wires attached to it. There should be a solid green wire on one gauge terminal and the other terminal should have the green/blue sending unit wire on it. Make a short jumper wire and connect it between ground and the gauge terminal where the green/blue wire is. If you turn the ignition on and the gauge goes to hot, that says there is a break somewhere along the green/blue wire going to the sender.
If the jumper wire does not make the gauge read hot, that says there is either a problem with the green wire supplying power to the to the gauge or a problem with the gauge itself. The solid green wire comes from the " I " terminal on the voltage stabilizer. Since the gas gauge works, that says the problem is not in the voltage stabilizer. You can test the green wire on the temperature gauge by setting a multimeter to measure Ohms and connecting its leads between the green wire on the fuel gauge and the green wire on the temperature gauge. You should measure something close to zero Ohms. If the meter shows an open circuit, replace the green wire to the fuel gauge. If there is low resistance, clean all the connections and try again. If there is still no movement in the gauge when you short the jumper wire to ground, replace the gauge.