Nice work, an edge beveler, to round the corners and a hand held sick type or plastic, or wood wheel can be hand held or mounted to a drill and some bees wax, hideglue or even white craft glue will give you a smooth finished edge by burnishing.
Interesting thing about the 'pinstripe' is that it doubly-establishes the edge of the strop.
If done properly, What is at the edge of the strop does not matter so much?