This simply means Barron isn’t a natural-born citizen, that Barron can’t run for president, but, I believe, Barron would still have inherited citizenship through his father via his mother’s approval.

Since his mother had the choice between naturalizing Barron into her home country or naturalizing Barron into the United States, inheriting citizenship from his father would have been a secondary act, more like adoption, whereas all “adopted” citizens are naturalized citizens.