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.
From an electrical point of view, NC pins need not to be soldered. Ripping it off or soldering it to a (small) pad makes no difference.
From a mechanical point of view, having it soldered or not can make a difference though. I am fairly sure that the manufacturer recommends a certain footprint in the datasheet, and that that footprint includes the pad for pin 10.
If you are confident that the other pins give enough mechanical stability, you can just cut it away as per your plans.
An even better option, from a manufacturing point of view, would be to just remove the pad for pin 10 from your PCB, and route the track from pin 9 right below it. The solder mask will guarantee electrical isolation, and there is no need to add a manual step to the manufacturing process. As Tom Carpenter as commented, there is a drawback: the solder mask thickness can prevent the chip to seat flush on the copper, thus leading to soldering issues in some cases.
Whether you trust your solder mask enough for this is a decision you should make.
The major reasons are already covered in other answers/comments, but also:
You risk damaging other pins, either physically or through electrostatic discharge.
In some cases, heat is dissipated through the pins. Modifying a pin can hinder this process.
It looks really unprofessional, and your customers/boss might care about that.