There are two questions here. The first is does pressure fall off more slowly in warm air than cold air? Imagine that pressure at sea level is uniform over a large area, but that the air to the north is colder than the air in the middle and the air to the south is warmer than the air in the middle. Uniform pressure at sea level means that there is the same total mass of air above all points we are thinking about. Since, at least at sea level, the pressure is the same everywhere, the cold air to the north will be densest and the warm air to the south will be least dense. So, a column of cold northern air, say 1 sq.m in area and 1,000m high, will weigh more than a similar column (the same volume) of warm southern air. At a height of 1000m above the sea, a greater proportion of the total mass of the atmosphere air will be below 1000m in the north than in the south. So, at 1000m, pressure will be lower in the north than in the south. So to someone going upwards from the surface, pressure will fall off faster with height in the column of cold air than in the column of warm air.
And is the fact that this makes higher winds more westerly due to the exaggeration of the effect of the earth's rotation on pressure flow? In my example, we have created a gradient of pressure, with lower pressure to the north and higher pressure to the south. On the scale we see on weather charts, wind flow is a balance between two forces: one is the drive on air to move at the same height from higher to lower pressure (just like air coming out from a bicycle tyre) and the other is the effect, as it seems to us on a rotating earth, of air being driven across its motion in direct proportion to its speed (to the right in the northern hemisphere and vice versa in the southern hemisphere).
You can try this "Coriolis effect" on a roundabout at your local rec: Take a cloth dripping with clean water. Get someone to set the roundabout spinning in the same direction that the earth spins (anticlockwise in the northern hemisphere). Then throw the cloth through the air over and across the roundabout so that drops fall on it but the cloth flies right across in a straight line. Then stop the roundabout: on it, the drops have fallen in a line which curves off to the right.
If it's wet outdoors, you can see the same effect by taking a piece of paper and holding it firmly on an old table top with a pencil between your lips and pressing down with your teeth (this pencil is the north pole). Then, with a second pencil in your right hand, draw a line going straight away from you at the same time as you rotate the piece of paper anticlockwise about your first (north pole) pencil. It's best to look into the distance and not at the paper. You've drawn a straight line but on the paper it is curved. Then do it again, rotating the paper clockwise about the pencil held in your mouth which becomes the south pole. Try varying, one at a time, the speed of rotation of the paper (variation with latitude on earth) and the speed you move the pencil in your right hand (wind speed). ]
The point is that there is no "exaggeration", just the balance between the two effects on a rotating body. In fact it is only when the forces do not exactly balance that we get vertical motion of the air and "weather".