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At the end of the problem video you can see the final shape has moved. When the path from the first shape was copied, it moved.
select the path and move it into place. make sure you move the path, not the position. you need to select the path property in order to do that.
it appears that I will have to manually move each vertex of the plane's path back where it came from individually, which is certain to be time consuming and error prone..surely this isn't necessary and that there is a better way?
of course not. you need to make sure the first vertex is in the same place. read here: http://www.provideocoalition.com/managing_moving_masks/ it's about masks but the behavior is the same for any path shape. mask interpolation is also for masks but if you need to, you can copy your shape path into a mask and use it, then revert back or not. more about first vertex here: Managing and animating shape paths and masks in After Effects
you can also use our Ae's community's guru Rick Gerard method of blending shapes in Ai using the blend tool and importing them as a sequence: Morph in Illustrator - YouTube
Did you solve your issue with Roei's response? Please let us know!
Thanks very much Roei Tzoref, this was definitely it. A little embarrassing to be honest - I thought I had already tried this but when I tried it again this fixed my problem. Thanks!