You have worked it out yourself. It is very common to use the heating element as a voltage divider to power the fan.
To make it run cold, you might get away with just a resistor, which would produce less heat than the heating element, as you would use it as series resistor not a voltage divider. However you would have to work out the value carefully.
A transformer feeding the motor, via a rectifier would work.
Either way you would need to measure the running voltage on the motor, with the usual care when working with mains. If you go for the resistor method, you also need to measure the motor current quite accurately.