In the US where they run on 110v how do they get on with cable sizes? I always understud that the lower the voltage the higher the current so their cables must be way larger than ours!?

The other problem with split phase is the need for double pole switching and fusing which is why it can be dangerous to use US 110 volt equipment on a British building site as our 110v is really 55-0-55 and needs double pole protection.
Heres one for you OP, when was the last time you saw a character in americian tv series or movie using an electric kettle? they all seem to have coffee makers instead!
At 16A and 110v I expect the reason they are rare over the pond is that no one would have the patience to wait for them to boil!![]()

The kettles seem to be either 5A or less designed for caravan use or around the 10A mark. I would agree the 3kW is rather rare now but they seem to jump from either 1.2kW to 2.3kW very few between the two sizes.
Because the voltage can vary from 220 to 240 often they are shown as 2.3 - 2.7kW or similar the element having a resistance of 21 ohms. This shows quite a large change in kW rating for what seems a small change in volts as the watts change is related to the square of the volts.
With 110 volt systems volt drop is more of a problem and with the kettle I have found with the low wattage versions designed for caravans getting the auto switch off to work with long power leads is a problem. So one would expect electric kettles for 110 volt may not have the auto switch off option.
Combine this all together and one can see why electric kettles are not so popular.
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