I'm tempted to provide the highest resolution icon and let iOS scale it down as necessary. Is this a bad idea, and (if so) why?
Yes, it might be bad idea. If your icon art was produced using a vector image editor (e.g., Adobe Illustrator, Adobe Photoshop with vector primitives, Sketch, etc) you best bet is to export all required resolutions yourself. In doing so, you will guarantee the best possible output for each resolution.
Keep in mind that exporting a single high resolution image and then resizing, for each required resolution, using a bitmap editor (e.g., macOS Preview) won't produce the same results.
PS. As already pointed out, I'm not sure your binary will even be accepted by Apple if missing icon data :-(