Well, here's the end of the story. I brought the car to the dealer, telling the service man that I'd found a p2181 code, I'd replaced the temperature sensor (his response: "good guess") and cleared the code, but the code came back.
They called about an hour later saying they'd fixed it. Dubious, I asked how: the answer was that they'd cleared the code. I explained that I'd already done that twice, and another time wouldn't help. So, they said the next thing to try was replacing the thermostat and coolant.
Still dubious, I asked if their scan had revealed more information than mine, and they said "no". I asked why they thought it was the thermostat and not something else, and they said that that usually fixed it. I asked what if it didn't, and they said the next thing would be the water pump. Continuing to be dubious, but not having much of a choice, I let them do what they suggested ($220, plus $100+ for the "diagnosis").
Well, I've had it back for two days now, with a fair amount of highway driving. I've watched the coolant temperature (I leave the Actron connected), and it does exactly what it's done all the time, with the exact same high temperature. Still no change in drivability, engine behavior, heater output, etc. But, there are no codes, active or pending. So, it looks like I'm finally fixed. Yee haw.
It's pretty annoying my engine computer will decide that something's wrong, but won't explain what the problem is even to the VW techs. In my mind there's a real possibility that this has been a wild goose chase, trying to find something that will pacify the engine computer, rather than fix a real problem with my car.