Does peeling skin around the nails, which reveals what is beneath it when lifted, constitute a barrier that prevents water from reaching [the skin] during Wudu? And must it be rubbed or peeled off permanently to allow water to reach what is beneath it?

These are baseless obsessions. You should pour water on your hands during ablution so that it reaches all parts of them without scraping the skin or removing flakes. This skin is considered part of the outer body, and washing it is sufficient. The default is that there is nothing that needs to be removed. Anything that cannot be removed from the skin or flakes without pain does not need to be removed. Therefore, you must abandon these obsessions and pay no attention to them.