coolant temp gauge would still show when key is turned off, that's normal
like here:

Or do you mean when you drive and engine is hot, the gauge does not register at all?

the sender would be somewhere here, on the thermostat housing

 

I'm a bit curious about this behavior. It looks like some temp gauges have a resting position in the middle, other (including mine) rests at min, like the gas and oil gauges. Can you please do the following test, with a cold engine:
1) switch on the power (no need to start the engine). Temp gauge should go to ambient temperature (around the minimum)
2) switch off the power

Does the needle stay at min or returns at middle? If it returns at middle, it means that the resting position is at middle for some odd reason.
But I bet that it stays at min, It means that the needle developed friction on its axis that prevents it returning to rest. This woud justify the difference between gauges (temperature gauge stays in the same position during driving, the others float).