UNreasonably Exhaustive Research: Finding a Mother & a Father for a Man of Common Name, Born c1784–88, Virginia

How badly do you want to solve a research problem? How many hours and strategies do you consider reasonable? How wide a geographic swath do you consider doable? William Mills “just appeared” in Franklin County, Virginia, in 1816, signing a bond for his planned marriage. At no point in his life did he associate with any person of the surname Mills, except his wife and children. He died 1860–64, leaving no probate. Death registrations were mandatory, but none exists for him. Others have plugged him into an unconnected family by merging him with a totally different, same-named man. Triangulated Y-DNA tests by descendants of three separate sons place him in no Mills line anywhere on the globe. Instead, his Y-DNA is that of a totally different surname found widely across Virginia but not in the county where he emerged as a grown man. How can we break through this kind of brick wall?