This is one of my "hobby horses".
While as long as the ratio of resistors is high, it makes little difference to the actual drive voltage to the FET, it is a matter of understanding the engineering principles.
It is not a matter of "needing to pull down the FET gate". The reason you have a pull-down resistor is that until (and unless) the Arduino code initialises, all its outputs are floating and the gate voltage would be otherwise undefined. The threefold consequences of this are that the device controlled by the FET may be prematurely activated to damaging effect, that if the FET is only partially switched on, it may overheat from the load current (hardly a problem here), and that in this particular case, the system will switch on spontaneously.
Actually, that last case is not true anyway as with no power to Vcc, the protection diodes in the Arduino chip do definitely pull the I/O pins down to 0.6 V or less and the pulldown is actually unnecessary.
So the pull-down resistor is there to address a problem generated by the Arduino, so should be connected to directly pull down the Arduino pin itself, not the FET gate.
Also where the pull-down resistor is necessary due to the driver circuit not actually having a totem pole output, such as where it is being driven only by a PNP transistor, the pulldown needs to be a much lower value to provide sufficiently fast switching.