The engine coolant temperature sensor on a vehicles engine is there to detect and know the engines coolant temp so that the computer will know how hot or cold the engine is.
The computer needs to know the temperature of the engine and the engine coolant so that the computer can tell the fuel injectors and the throttle body and other related components how much fuel and air mixture the engine needs to run properly while it's either cold or either warmed up.
When your engine coolant temperature sensor is not working properly or has completely failed then your engine will be hard to start when it's cold and it will stall when you try to drive off.
If your vehicle is experiencing rough starting or rough idling or even stalls after taking off before the engine is fully warmed up then it could either be a bad coolant temperature sensor or even the connection to the coolant temperature sensor could be corroded or may have come loose as I've seen some coolant temperature sensor connectors become loose or sometimes not even attached.