What if I cut off an NC pin from an IC?
Sometimes "No Connect" means there is no connection between the package pin/pad and the IC die. In that case you can route whatever you want through the corresponding PCB land. Sometimes "No Connect" means "thou shalt not connect anything to this pin", because it is connected to something on the die but reserved for factory test or some other purpose. Sometimes the datasheet makes it clear which of these applies, sometimes not - from the text you've quoted it's not 100% evident but points in the direction of the latter, which is always a safe assumption to make.

In this application the trace impedance is going to be trivial compared to the gate drive IC's 1+ ohm output resistance, so you really have nothing to gain by changing the layout from your present design.