Does the Stern-Gerlach experiment really suggest superposition? The Stern-Gerlach experiment is often cited as evidence of quantum superposition and there are some very simple explanations like this one https://www.youtube.com/watch?v=hkmoZ8e5Qn0  in which spin on one axis is compared to colour and spin on a perpendicular axis is compared to hardness.
The Stern Gerlach experiment relies on the inherent angular momentum of a silver atom giving it the properties of a tiny bar magnet.  When a beam of particles are fired through a large external magnet field, half of them are deflected up and half are deflected down, with respect to the orientation of the external field. This does not surprise me at all. I am reminded of a high school experiment in which iron filings sprinkled onto a piece of paper which is placed over a bar magnet to demonstrate the lines of force. I imagine something similar happening when a beam of atoms are fired through the Stern Gerlach apparatus; the particles align themselves with an external (UP/DOWN) magnetic field as they pass through it. $50$% go up and $50$% go down. When only the UP aligned particles exiting the apparatus are directed through a second apparatus, at right angles to the first apparatus (LEFT/RIGHT), $50$% go left and $50$% go right (that is up or down with respect to the alignment of the second apparatus).  This is no surprise to me at all because I imagine them simply realigning themselves to the magnetic field.
When ALL of the particles exiting the second apparatus (which are now aligned either left or right) are directed through a third apparatus, with the same orientation as the first (UP/DOWN), they ALL come out aligned UP. Admittedly this seems odd, but perhaps they have retained enough of their UP alignment to be forced back into an UP state.
In short, it looks like the the orientation of the external magnet field changes the orientation of the particles.  Why is this evidence of superposition? Does it not simply suggest that spin is changeable?
 A: The last step is incorrect. The sequence is

*

*Atoms are split into $|\uparrow \rangle$ and $|\downarrow \rangle$, but only the $|\uparrow \rangle$-atoms are kept (=atoms possessing a spin which points in the $|\uparrow \rangle$ direction).

*Atoms are split into $|\leftarrow \rangle$  and $|\rightarrow \rangle$, but only the $|\leftarrow \rangle$ atoms are kept. Thus, one would expect that the spin of the atoms points $|\uparrow\rangle$ and $|\leftarrow \rangle$.

*If these atoms are again brought into a B-field which separates the atoms into $|\uparrow \rangle$ and $|\downarrow \rangle$, one would expect that we obtain only $|\uparrow \rangle$ atoms, because we kept only those atoms in step 1. However, we find that the atoms are again split into $|\uparrow \rangle$ and $|\downarrow \rangle$. Thus, we conclude that the second measurement scrambles the $|\uparrow \downarrow \rangle$ spin direction.

I wouldn't say that the Stern-Gerlach experiment is widely used to show the superposition principle. The superposition principle is not surprising in physics, because it is standard for linear systems. Instead, the Stern-Gerlach experiment shows that the two spin measurements (along $z$ and $y$) do not commutate: This means that the order of the measurement matters. Although this is not a big deal in every day life (e.g. it matters if you first open the window and then put out your head, or visa versa), this was a big deal in physics.
Minor detail: The magnetic field must be inhomogeneous. Else the atoms would not experience a force, because $F = \nabla E = \nabla (-\mu\cdot B_z) = -\mu \cdot \nabla B_z = 0$, if $B_z=const.$
Edit: I didn't watch the video before. Now that I watched part of the video I realise that your question is not about the original Stern-Gerlach experiment (sidemark: Please try to post a self containing question). The missing twist in your question is that step 2 is performed (the atoms are split into two components), but these components are not measured. Instead, the two components are recombined -- note the "mirrors" outside the "Hardness Box" (=spin measurement in $y$-direction) in the following image (taken from the linked video)

After recombining the two path we obtain the same result as if the "Hardness Box" were removed, because no measurement was taken. So, yes, in this context the experiment shows the superposition principle of pure states, because the result only makes sense if we argue that the atom has taken both path simultaneously. Only if this is the case no measurement was performed. However, I feel that this point is clearer in the quantum eraser experiment, which uses linear polarisers and the standard Young double slit setup, or other "which way" experiments.
