I think it should be viewed as a chain of infinitely increasing bevel gears, each gear representing a local deformation of the shaft, allowing for a transition from discrete to continuous view.
For the system to work, the assembly must be rigidly held when the shaft is rotated, as seen in the video (two attachment points at 90°, with the end free to rotate). Without this arrangement, attempting to rotate the shaft would cause the assembly to rotate. If we include these holding forces, we can conclude that the change in orientation of the shaft does not change the angular momentum of the entire assembly.
What is misleading is viewing only the flexible shaft as an isolated system, when it is not; the support must be included in the analysis.
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