There has been a bit of progress, which I want to update the "world".
On the discussion between the managers of the Geni profiles, someone helped me locate the Dorot HaRishonim volumes on line. The actual page link of the above is:-
https://www.hebrewbooks.org/pdfpager.aspx?req=20282&st=&pgn...
(and the page after).
The style of Hebrew does not make it easy reading, at least for myself, but one matter is very clear. He talks about the lines of succession with respect to the academies, but not every successor was the son of who was being succeeded. In some of the cases he mentions that "a" was a son or grandson or descendant of "b" but not in every case. So, for example:-
In Pumpedita, Rav Meri the kohen from N'har P'kod (The River P'kod) succeeded Rav Shmuel in year 4511 (in civil calendar this is either 751CE or 750CE depending on whether before or after Rosh HaShanna) and was head for eight years. At least not here, are we told who the father of Meri the kohen was, and obviously it was not Shmuel if he was not a kohen. He does mentions Shmuel's father as Meri, but this is obviously a different Meri who was not a kohen.
On these pages, he does not seem to get into the issue of Davidic lineage, but also I might have missed this. It is very likely he mentions this in other parts of his works.
His sources seem to be Raba"d, but am not sure whether it is: https://en.wikipedia.org/wiki/Abraham_ben_David or one of the other rishonim (Rabbinic scholars who succeeded the ga'onim) who had this abbreviation as their pen-name. (See article there for further references) and Rasa"g pen-name abbreviation for Rav Sherira Gaon https://en.wikipedia.org/wiki/Sherira_Gaon . He understood that Raba"d had errors that he would not have had, if he had access to the Rasa"g's writings.
I have already sent this response (slightly modified) to the managers of the profiles and also will try and reach out to those who are busy with the relevant Wikipedia articles that I previously mentioned, stating that their "source" does not give this information and the geni trees that they also reference, seem to be based on the same misconception.
Regards,
David