Hmm. I agree with the examples in #1 and #2, and I suppose the singular noun Familienmitglied would be one way to translate them.
However, I don't really think it explains the underlying syntax, because I don't think the English word is really being used as a noun. It feels to me more like an adjective, because after all, it's not countable. You could never use the indefinite article; we do not say 'She's *a family.'
So to understand the English syntax, you might need to make up an adjective like 'familienverwandt,' or else use a paraphrase like 'sie ist Teil der Familie' or 'sie gehört der Familie.'
And I'm not actually sure I even approve of the tweet in the opening post, which probably should have read 'she was the first relative / the first family member he called.' Again, I don't think I would use 'family' in this sense with an article at all. To me, the first family is 'die erste Familie' (bzw. the First Family / die Erste Familie meaning the occupants of the White House).
Not sure if that helps, but fwiw.
(OT: BTW, I can only imagine that the queen's heart probably sank at the use of her very informal childhood name as a formal given name, not to mention the pairing of it with the name of a daughter-in-law from whom she had been estranged. Why they didn't just have the good sense to name the child Lily and say that it reminded them of Elizabeth, I can't imagine.)