Sadly, nobody has confirmed (or otherwise) my attempt to estimate of the time expansion ratio of the movie clip.
The interesting fact about the strut deflection is that it occurs at a frequency that is roughly double that of the kerbing input. I have struggled to find an explanation for that. On reflection, I think that the "buckling" reference I found could well contain an important clue, under the heading
"Flutter instability", although it doesn't look very likely at first sight.
I ask you to imagine that the two beams of the "Zeigler column" are the simplest lumped parameter approximation of the push rod, and the outer ball joint is represented by the point "C" in the diagram. The outer ball joint must be assumed to have, what is called "dry friction", as I suggested (kind of) in a
previous post. Now, with that in mind, you might want to watch the explanation contained in this
video, also referenced in the "Flutter instability" section.
If you find that convincing, then the solution would be to reduce friction in the outer ball joint. Alternatively, an FE model might be used to confirm the instability(*), and to investigate other changes to improve the "stability margin" of the strut. If I were to do that, I would very much like to know the actual frequency of the instability.
Overall, I think that JT's conclusion was correct but, perhaps, the reasons are rather more complex than would appear at first sight.
(*)Edit. Incidentally, I would include the upright in the model, & also the power steering system (I guess it would be hydraulic).