Good news! My attempts with wearing the fitbit in reverse, i.e. with the sensor at the underside of my arm, were successful:
This time the fitbit alta HR and the Polar H10 agreed much more closely, even during an hour of running! It also correctly detected the drop in heart rate in the middle of my run when I stopped for a bit.
So while not accurate enough for serious heart rate tracking during exercise (for that I still recommend a chest strap like the Polar H10), the fitbit is definitely good enough for 24/7 monitoring and more casual tracking during exercise.