I'm not sure how hot the actual filament will be surrounded by insulation and water presumably at 100°C.Tell me how hot the heating elements will be 'after the initial few minutes' and I'll do the approximate sums for you
It could be much more than 100°C.
They are according to the measurements, e.g. immersions are 3kW and around 19Ω when cold.The other crucial question is, of course, whether you are right in saying that a '3kW appliance' is 3kW 'when cold' - or whether the 3kW actually relates to 'after the first few minutes'.
Lamps are differentAfter all, with filament lamps, we quote the wattage when hot, not cold (the latter of which would be extremely high, for at least a few milliseconds).