It is not only a matter of a current dividing between the two. The voltage drop across will product a voltage to apply to the pin. This may be to provide an adequate input signal to that pin.
I don't know the resistance of pin 8 when the circuit is active. I couldn't find a schematic for the Uno board.
I do know that in some of the projects later they take the output from the resistor and apply that directly but they don't seem to have a resistor in series with it. If they do you can assume that is the resistor is turned all the way to one end it would provide a short and then just short the center to the voltage that would be present if the 10K were 0. I did this on the Lcd projects as it was only used to adjust the brightness. I also didn't have to press down on it all the time.
It's called a pull-down resistor. It pulls the voltage on the pin to ground (logic '0') when the switch is open. Without it, the pin voltage would be floating when the switch is open. A 10Kohm resistor is used because too small a resistance would draw an unnecessarily large amount of current when the switch was closed.