The entire point is that any horizontal temperature gradient will produce separated regions of warm and cold air which necessarily have differences in density that then produces a PRESSURE GRADIENT. All pressure gradients have to be balanced by geostrophic flow and so a wind develops and increases with increasing height as the pressure differences between region A and region B in the diagram above, INCREASE with height.

Ultimate it is this horizontal temperarture gradient which produces differential presssure between warm and cold air that increases with height which then powers the Jet Stream.