In structural engineering, a pad footing under a normal load N and a moment M is assumed to have a linear distribution of stress under it. Is there any intuition why we can make this assumption?
The stress distribution must be in equilibrium with the applied load, and it is a reasonable assumption that it cannot be in tension. Notice how that creates the difference between cases (b) and (c) in your attachment, depending on the relative size of the direct load and the moment.
There must be at least two independent parameters in the stress distribution to match up the two independent parameters in the load (i.e. the direct load and the moment).
In the general situation there is not enough information to determine more than two parameters, so if the simplest assumption (a linear distribution of stress) is reasonably consistent with experiment, there is no reason to do anything more complicated.
A more complicated assumption with more parameters would require more assumptions about the behavior of the footing.
Civil engineering is not an exact science. If simple assumptions plus a safety factor are enough to make designs that don't fall down when you build them, that is usually "good enough" unless you are designing a safety-critical structure.
Soils strata deemed being competent to bear load have elasticity and a measurable linear young modulus of elasticity unless verified by in-situ tests that the underlying layers are of different properties by a soils engineer.
Elastic modulus can be estimated using SPT 'N' value or based on static cone penetration resistance (qc)
Sand (Normally consolidated) E = 500(N+15) or E = 2 to 4qc
sand(saturated) E = 250(N+15), sand(over consolidated), E = 6 to 30qc
gravelly sand E = 1200(N+6)
clayey sand E=320(N+15)
silty sand E=300(N+6)
if the site is critical or the building authority requires a geotechnical report then the report will have that values.
And any elastic materials strain linearly and your sections are valid.
However if the geologist report indicates different layers of soil present at the job site, it has to show the contour lines and daylight line, and bedding angle.
Then the engineering in coordination with the soils engineer will come up with a design scheme for that particular project.