The circuit being "live" is a safety thing - do you turn off your high power electric circuits by switching the ground off? Or do you take away the 120V source instead?
If you look at sheet 2 of the part you found, you'll see thisGate-Source Leakage IGSS VDS = 0 V, VGS = ± 8 V ± 100 nASo not much current is going to flow into the gate.
When the gate switches on/off tho, you want it to do it quickly so you don't have 5A trying to flow while the part is transitioning from off to on.
A lower value resistor to pull the gate high quickly to turn off quickly is thus better.