I managed to locate the engine temperature sensor and was able to pull off the connector. Put the car into on/run mode and plugged in the RaceCapture Track mk2. Got an engine temperature reading of 419 F! Note that this is 215 C and the PID conversion is "value - 40", which is means that value = 255, which is the max value of an unsigned 8-bit number. So, this confirms to be that the engine temperature sensor is working. Good thing I don't have to replace it because there is almost no room to work with there. I believe I could swap the sensor without pulling the intake manifold, but it would require a 6-pack and inventing 37 new swears to pull it off.
So, I'm leaning towards the "something else is going on"... and I'm starting to think that this might be an electrical issue.
I know the fans run when the A/C is on. I started up the DD, which is an Impreza and put the A/C on. This started the radiator fans and they shut off when I shut off the A/C, so working as expected. However, the fans were running pretty slow, whereas the fans on my car were on full blast. I could take a video of them, but the best thing in my head to compare it to would be a box fan - the Impreza is on low setting, whereas the BRZ is on high setting. Not sure this totally discounts the theory that the BRZ thinks the A/C is actually on, but the behavior seems different enough.