Intuitive "story" explaining how orientation of spin axis affects up/down observation? Is there a "convenient fiction" that explains why the angle of an electron's spin axis affects the probability of it being observed in a spin up or spin down state?
By "convenient fiction", I mean a story or image that provides useful intuition to novices, even though it may not be technically accurate.  For example the analogy of water flowing through a pipe is a convenient fiction used to introduce the concepts of current and voltage.
I imagine the electron being sent through a Stern-Gerlach device.  It makes sense that the closer the spin axis is to vertical, the more strongly the electron is drawn up or down.  But, I don't see what would induce the electron to ever move in the "unexpected" direction.  For example, if the axis is 5 degrees off vertical, what ever induces it to move down?
Watching this Veritasium2 video leads me to imagine that the electron is constantly flipping its spin axis; but, that doesn't seem to explain how the angle of the axis affects the probability of being measured in the up or down position.
https://www.youtube.com/watch?v=v1_-LsQLwkA&t=334s
 A: I am afraid there is no way to get an intuition. The Stern-Gerlach experiment implies a description of the reality in terms of a superposition of states. In case of the electron spin you have Up and Down states related to the direction in which you measure. A surrogate of intuition could be the coefficients of the base kets (base states) specifying the superposition. In fact the probability of finding the electron Up or Down is the modulus squared of the related coefficient.  
However the image that the electron moves up or down is not justified. In quantum mechanics you speak of collapse of the wave function during the measurement process.  
A: My preferred visual (which, yes, is a convenient fiction) is a two-part image.
In the initial image, I visualize the same picture you seem to be using. That is, I think of the electron like a billiard ball spinning around an axis that is literally pointing in a certain direction. In the $5^{\circ}$ off vertical example you wrote above, I'll see the electron's spin axis superimposed on the standard $\hat{z}$ axis, tilted by $5^{\circ}$.
But then, like you, I get stuck with "but how does that give it a chance to point down?" I remember the $|+\rangle_z$ and $|-\rangle_z$ states are orthogonal, so I say "okay, instead of $|-\rangle_z$ pointing in the $-\hat{z}$ direction, it's pointing in the $\hat{x}$ direction." At this point, the basic rules of geometric projection come into play, and are clearly present in the picture in my head.
Of course it doesn't make much literal sense to do this -- in real space, the $|-\rangle_z$ direction is not along the $\hat{x}$ axis -- but that's the downside of looking for intuitive visuals in physics :) If you're like me and this is the way you do physics, you probably already know that it helps to have a collection of such convenient fictions for the standard systems, and to know the ways each of them fails to capture the behavior of the true system or, equivalently, in which contexts each visual happens to work. (And to always check your intuitions with the math!)
