As you know the ECT varies resistance with temperature. The higher the temperature the lower the resistance. Correspondingly the higher the temperature the lower the voltage signal to the PCM. Higher temps will signal PCM to give leaner mixture. I'm doubting the fail-safe is a warm engine signal to the PCM, otherwise the problem would be hard-starting/stalling at cold start-up. I still think the 30-60sec problem window is too narrow to assume an ECT problem, they just don't react that fast and engine temp would have hardly moved.
You can test the ECT with an OHM meter and compare to specs or test voltage signal to see what voltage is being sent to PCM, but good ECT ground would also affect voltage, so you'd need to test that as well. As well if the reference voltage (5v) is low then the reading will be off. Some PCMs seem to have an internal "pull-down" resistor that reduces signal voltage further.
What may also be occuring is overnight leakage of an injector or two that creates a flooded condition in one or more cylinders. That flooded condition is something that will correct in 30-60 sec. A backyard DIY could pull all the plugs before letting the truck sit overnight so that any leakage would evaporate during the night. Install the plugs just before cold start and see if the rough idle is still there, if not then that's probably the problem. You could also get a used oil analysis which would show excess fuel in the oil if overnight leakage, assuming piston blow-by is minimal.