Why are the planets orbiting the Sun in a flat disk, instead of in all different orientations like the popular depiction of an atom?
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You have the problem reversed (should be “why aren’t electrons arranged like a planetary system?”). A fluid, when rotated, will see material migrate outward from the center and flattening of the sphere into a disc (the distance along the axis shortens as material migrates outward near the equator). The fastest motion is occurring at the equator (most distance traveled per unit time the further away from the axis of rotation the matter is located) so that stuff is impelled to move outward even further. The slowest motion is happening at the axis (zero displacement at the very axis), so that material is “pulled” parallel to the axis and toward the equator in order to replace the lost materials in that region.
Liquids, gases, and semi-fluid mixtures will all do this sort of thing, whether water on a spinning plate or clay on a potter’s wheel. The main concern is what other forces or processes are affecting the way the fluid responds over time to that push to spread outward along the equator and shorten the axis of rotation?
With planetary systems, the center is a strong gravitational attractor, which is what keeps the disc from becoming depleted in the center zones with time. When gravitational attraction is powerful, the outward movement is largely offset by the pull back toward center. You end up with an “oblate spheroid” like the earth, where the diameter at the equator is longer than the diameter along the axis, by some percent depending on rate of rotation (with no rotation, there is no outward force to offset and the distance from center of mass will be the same at surface everywhere, because gravitational attraction is an inverse function of distance ONLY in that situation).
The sphere is flattened, and it is just a matter of how much “cohesiveness” keeps it from distorting into a disc rather than a flat-ended sphere (a disc is basically just a sphere which really flattened out). The solar system is pretty diffuse (not much mass per unit volume and no other important forces acting at the scale of the system) so gravity is not driving the retention of a spherical shape.
In atoms, there are multiple forces involved, with electronic (charge) attraction and repulsion dominating. This forces the electrons into movement patterns that maximize the attraction toward the nucleus while minimizing repulsion by interaction with nearby passing electrons. You end up with a cloud of electrons, with each electron traveling in high-frequency zones and not drifting (much) into other regions occupied by a different electron where that region is its high-frequency zone.
The mathematical description of the various forces that come into play is somewhat complex, but the solutions to the applicable equations indicates these oddly shaped “orbitals” or “electron paths” provide the minimum energy conditions (least repulsion or other interactions with maximum attraction to keep the electrons in the zone of influence of the positively-charged nucleus). Spinning rate (displacement speed) is actually not variable (unlike with planets, solar systems, water on a plate, clay on a potter’s wheel), given that electrons are basically all moving at the speed of light wherever they exist in the “sphere” of the atom. The rotation of the atom is not what is driving the movement.
We do see evidence that the predicted electron zones are actually and truly occupied, via the way that bonds form between different elements and other things. But we really do not “see” where the electrons are truly located specifically. We calculate that distribution. It is not a “popular” depiction. It is what we believe is a good way to represent the probability distribution for the location of given electrons. Most people only know the basic structure expected for a low-number element anyway (first two electrons of the S occupy a spherical zone and the next 6 p electrons are located as “dumbells” along the x-y-z axes. The weird donuts and disconnected spots of higher number electrons are generally not known.
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