The Hall current effect is merely due to the sideways magnetic force on the drafting free charges. It becomes important with magnetohydrodynamics applications, as it acts towards a strong magnetic field (so that the influence of electromagnetic force is noticeable) and towards a low density of the gas (such as in space flight and in nuclear fusion research). With these specific applications, you are right that it is often therefore overlooked.
The effect of Hall current gives rise to a force in the z-direction, which induces a cross flow in that direction, and hence causes the flow to become three-dimensional.