The rotating point of view was the "aha" for me. It's even clearer to me when you look at the 1:1 ratio with two identical coins, where when the rotating coin is upright at the bottom of the rotation, and the coin is right-side-up to the observer but is "top edge to middle coin" instead of "bottom edge to middle coin".

(Specifically, my "aha" was that one rotation of the moving coin in place was not the same as one rotation of the moving coin around the middle coin.)