Imagine you have a steel column as a solid square of 6-inch by 6-inch section carrying a load P sitting on a square base plate, 12-inch by 12-inch.
The normal stress on the base plate is not $ \ \sigma= \frac {P}{12^2} $
The stress due to the P on the baseplate will be a complex mixture of normal stress, shear stress and bending (moment) stress.
if we consider just the normal stress on the baseplate it follows a pattern of concentric roundish square contour lines which are degrading in intensity as they open farther.
I attach a crude diagram here, approximately showing the spread of stress contours.

That is why we consider the stress locally just for a point. In this case, we want to make sure the maximum normal stress is below the allowed stress, not the average stress.
The distribution of normal stress into the depth of the plate is a bit complicates but basically the stress opens into the depth following an exponential spiral profile.
In many of the structural members like beams, stress increases linearly vertically as the point is farther from neutral axis of the beam. There are however structural members much more sophisticated than a simple beam where the stress varies greatly on a different point of a section, eg, cranck shaft of a car engine.
Some members have almost uniform stress load across their cross-section, eg, a truss tension member at a point not very near its joint.