So according to that theory there is no need to actually increase engine speed as it doesn't do anything?
It's not a theory, it's a fact. Net thrust produced by a turbojet engine is what's left over when the drag produced by that engine is deducted. Take away the thrust, and all you're left with is drag. Reverse is the act of taking away some or most of that thrust.
Spooling the engine increases drag, and thus the retarding force to which we refer as "reverse thrust."
While some retarding force in reverse is actually owing to reverse gas flow, in some aircraft, very little of that gas flow accounts for the acceleration (or deceleration, if you will) of the airplane. What's slowing the airplane is drag; this drag is highest at high engine speeds and high airspeeds.
Increasing the engine speed does everything. It increases induction drag, also referred-to as inlet or intake drag.