Oh, the 2.8 seconds is just a furfy, "a few seconds" is close enough! If the temp gauge is reading high a few seconds after a cold start then something is interferring with the temp sender's signal. If you tap the throttle and that goes away, there's a chance that there's an issue with the throttle connector/CAN line. It points to a possible issue with the ECU's connector, which given the covering on it makes you wonder if the engine bay's been pressure-washed.
If the temperature slowly starts to rise idling the car without touching the throttle, it means that the temperature sensor circuit is doing what it's supposed to and we're still looking at interference by the throttle signal.
The throttle signal is not fed to the instrument cluster, it enters the loom heading to the engine bay and up to the ECU, so it's not a direct interference (if it's interference at all). The common ground between the two is that ECU, its connector and its earth. Engine temps are fed directly to the ECU and the processed signal is sent in to the instrument cluster.
Querying the ECU for the temperature is going to tell you if the problem exists as an intermittent sensor failure (the ECU's coolant temp value drops when the accelerator is pressed) or if the problem is between the ECU and instrument cluster (pointing more to a instrumentation problem if the ECU is still reporting the coolant temps properly even when the gauge drops).