In a circuit with a switch and a pull-down resistor, why does the high impedance of the input not discourage current from flowing into pin?
Yes, all of the current that flows goes through that resistor to ground.  That way almost all the voltage drops across that one resistor and you are able to read the voltage with the pin on the high side of it.  Only a very very small amount of current flows into the pin when you read a voltage.  It just needs to charge up a very small capacitor.