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Looks like Epson has addressed this issue and has reinforced the rod in new generations of this scanner.
I've got the problem described in this repair, but when I stripped it apart I found that the end of the rod in question is thicker and seems to be less prone to splaying.
This rod connects directly to the Mode Selection switch (allowing you to toggle between sheet and booklet scanning.) When I engage the toggle switch for standard sheet scanning and try to force the rollers to move, I found the clicking happens at the cog connecting the rollers to what appears to be another cog designed not to move in order to stop the rollers. It seems these cogs are slipping while rolling instead of holding in place.
The only way I've found to stop this is to press down on the toggle switch to engage the gears more tightly. Not sure if there is a way to do it otherwise.