But the most important reason that airports publish air pressure is that they are used to adjust the altitude gauge so that pilots know their relative altitude above terrain with reasonable accuracy.
That may be true in non-US airspaces (I don't know my QNFs from the other QN...), but in US airspaces the altimeter in an aircraft is calibrated relative to mean sea level, not above ground level. All obstacle, terrain elevations, and instrument chart altitudes for non-military operations are given in height relative to sea level, not height above the ground.
As you pointed out, the barometric pressure is always given by ATC so that pilots dial in or confirm the air pressure setting into the on-board altimeter. This serves two purposes, namely that a) everyone talking to ATC all have the same altitude reference and b) pilots can accurately remain clear of man-made and natural obstacles, as well as properly fly instrument approaches.
