The EGT excursion is the result of many factors, but one has nothing to do with the airspeed increasing:
If you've been at idle (or near idle, as in a long taxi) the rotor and casings have stabilized at a moderate temperature. Now pour in the fuel for takeoff, and the first thing that happens is the engine temporarily is at a rich fuel/air ratio to develop the torque to wind it up to speed. Once on-speed, the fuel flow drops back to a value determined by the governor (which may be either mechanical or electronic).
But now the lightweight casing of the engine starts heating up and expanding, meaning that the rotor-to-stator clearances increase, meaning the gas leakage past the blade tips is greater. This inefficiency calls for a bit more fuel, and more EGT.
Finally the massive rotor starts to catch up, and as it heats, the clearances once again close down to a normal stabilized level. This restores the design efficiency, not as much fuel is required to hold RPM, and the EGT cools down a bit.
All this can be observed in a static runup. Add in the airspeed increase during takeoff, and this is exaggerated.
OK?