"Which is a bit bizarre, because they say it's better than, for example, the HRM on the machines at the gym, which measure continuously."
They don't measure continuously, fitness machines only measure when you have a firm grip on the electrical contacts and that's not possible during vigorous movement.
The one in the gym says my heart (just after running at 16km/hr, I can't hold the contacts at full sprint) is around 100, measuring it myself after the sprint puts it at 140+. My Fitbit is not accurate I suspect, it puts me at 130, but its a damn site more useful than the fitness machine's one and probably closer than my attempt to measure my pulse while on a treadmill at brisk walk.
Nuisance lawsuit, I wonder whose funding it.