I don’t know if I agree with this. Most Roman subjects during the Roman Empire did not consider themselves ethnically Roman in the sense we use ethnicity. While some people from some remote cultures may have fully assimilated into a more uniform Romaness over the generations, Rome was always a multi-cultural empire that retained distinctions between what we would now call ethnic groups as subjects. Just like Asian Americans, African Americans, European Americans, etc are all “Americans” we don’t consider them the same ethnicity even if they have been in the US for centuries. David Duke and Angela Davis are both Americans (and both named after the same, Jewish, biblical figure) but I’d be surprised if anyone said that they shared an ethnicity.
The second paragraph is not consistent with the rabbinic tradition of Judaism, which, before Zionism, was more simply called “Judaism.” King David’s grandmother was a convert. Converts have always been a part of the religion and the Jewish religion makes very little distinction between a person who was born Jewish and a person converted. And the child of two converts is just as Jewish as King David. Even the Convert is just as Jewish as any other Jew despite having no Jewish bloodline. I linked a podcast episode in another comment chain here that goes into much more detail and cites historical sources for these claims (that particular podcaster is much more qualified to speak on this issue than me).