It is simpler to think of the thermodynamic definition of temperature, to start with, the one on the left:
where N is the number of molecules, n the number of moles, R the gas constant, and k the Boltzmann constant.
For the expression on the right, randomness is implicit, Brownian motion after all led the molecular model, and it is basically random.
The identification of temperature as the same quantity in both expressions holds when the conditions for both expressions hold.
Moving a fixed volume at a fixed pressure does not change its temperature , it is a definite macrostate for which the microstates on the right are used to average out the molecular velocities.
If randomness does not hold , as with a moving system, the fact that we do not observe a rise in temperature in moving systems leads to an answer by Deechit Poudel, the rms of the velocity of gas molecules is large, and the contribution of the motion is small for every day motions. This can be verified with this calculator:
For 20C in air the most probable speed of the molecules is 400m/second, Larger than the velocity of sound.
For supersonic velocities and velocities commensurate with the speed of sound, the relationship demands as an axiom the randomness, imo.