“Then why the emphasis on genealogies in the Bible?”
There are certain promises and obligations of each tribe. For example, Levites and, as a subset, the Cohenim, have an obligation to serve in the Temple. This is a paternal-line obligation.
Similarly, from the line (of most recently) David comes the Messiah.
I have heard it expressed that “converts” are not actually “converts” but Jewish people born to other families, for whatever reason. The deal was struck at Mt. Sinai with all future, present, and past Jewish souls.
Jewish souls finding their way back after what may or have not been a difficult geneological journey.