I see that you invoke Biblical genealogies. Both Jesus’ maternal and paternal genealogy is given. I should tie down when and why the rabbis decided upon maternal lineage: it was in the second century AD. There are two reasons why: First, rape and such, from the time of the captivity to the dispersion of the Jews following the fall of Jerusalem, Jews became concerned about the implications of insisting on paternal lineage. Maternal lineage was at least uncontroversial. Second, it is that the culture of Judaism is more transmitted by the housekeeper than by the breadwinner.
I have done a lot of research on this and I am still on the fence on this one as there is evidence of this going back long before the 2nd century. Now it may not have been written down before then, however, there is evidence of it being practiced. I tend to lean towards the fact that is the mother who has the first influence in raising a child and therefore, if the mother is a practitioner of the Talmud, then the children will also be raised as practitioners of the Talmud, following every man-made burdensome doctrine of the rabbis.