Does the Stern-Gerlach experiment really suggest superposition?

The last step is incorrect. The sequence is

1. 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).
2. 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 $|uparrowrangle$ and $|leftarrow rangle$.
3. 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 $F = nabla E = nabla (-mucdot B_z) = -mu cdot nabla B_z = 0$, 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.