I think the easiest solution would be to take a precision bit you don’t need, and attach that directly to a 1/4” hex bit with something like J-B Weld SteelStik.
Short of getting the screws from HTC directly, the best option I could think of would be going to a hardware store to have someone size them and either replace them, or give you data to shop around for.
There’s GripStick Headphone Plug Extraction Tool for 3.5mm headphone plugs, but I’ve never seen anything smaller. Maybe you could create a smaller version yourself? If not, you could try putting CA glue on the end of a toothpick to extract it.
It sounds like the issue may be with the cable itself — have you tried other cables? Also, does your iPhone charge while powered on and connected to a wall outlet?
