Further, it seems as though, for any given starting point, you should be able to use this loop to determine the shortest traditional (non-returning) route, by eliminating the longer of the two distances attached to whichever starting point you chose.

My intuition suggests this may not be true, but I'm not a graph theorist and my intuition isn't trustworthy in these sorts of problems. But my thinking is this: finding a short loop seems to nudge the solution in a direction so that the distances attached to the starting point are short, whereas finding a short path from said starting point only cares about the distance to the next city and not the distance from the starting point to the final city. As an illustration, consider the numbers that I used. Any short path starting at A, B, C, or F ends with DE or ED, I believe, because D and E are close to each other and far from everyone else, which themselves are close to each other. But if I took a short loop, either I could find a consecutive triple of guys from A, B, C, and F, or I would have D and E are separated by two points in any direction. In the former case, say I had the string ABC, then starting at B and deleting BA or BC would not give me the optimal path from B. In the latter case, D and E are not consecutive, which should always happen if I start at A, B, C, or F.