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Next day, I went to the hardware store, bought some door hinges, as I saw recommended in a homemade-snow-shoe-binding-into-cross-country-ski-bindings retrofitting article.
I elected to use a narrow wooden plank as a base to which to attach the boot, to avoid side-to-side play.
Drilled out holes in the wood, and passed the same paracord through them, which held much better thanks to the "foundation" provided by the wood - I wove together a harness for my boot, and it held pretty well.
This worked a lot better, and I made it about 300 feet before they loosened up and fell apart, leading to yet another fall.