Coolant temp gauge problem

If the gauge needle sticks at off/home, a slowly climbing resistance may not cause movement. A zero ohm (short gauge to ground) can cause a gauge to do things that a within range (80-10) resistance will not. Shorting the sender wire to ground is a bit "shade tree" test but... everybody does it initially.
With only the vehicle for testing, you could remove the blue wire from the fuel gauge and put the purple temp wire there. If wiring is good your 40 ohm running temp resistance should move a good fuel gauge needle to just over 1/4 tank.