Jesus' David descent seems to have been commonly accepted during his life (Mk. 10.47ff.), and is attested also by Paul (Rom. 1.3). Hebrews 7.14 assumes that everyone knows Jesus belonged to the tribe of Judah.

As to the appropriateness of tracing Jesus' lineage through Joseph, who was not his biological dad, there was reason to show that Joseph was an heir of David's throne. Though Jesus was born without Joseph's, um, input, so to speak, he was still considered to be "of Joseph," and therefore "of David," because in every legal sense of Jewish life, Joseph was Jesus' father [i]in a legal sense[/i]. [b]Kiddushin 4.1[/b] states that a man and wife who live together are presumed to produce legitimate children without the necessity of proof of parentage. Since Joseph and Mary were legal married at the time of Jesus' birth, Jesus was considered to be Joseph's legal son and not an adopted one. Therefore Joseph, as Jesus' [i]legal[/i] father, would confer the heritage of David's line to Jesus. Even adoption (though Jesus was not adopted by Joseph) established one in the royal line, and adoptive sons of rulers would be called their sons.

Why is called the genealogy of jesus when it's about Joseph who isn't even his real dad? It'd be like someone writing the genealogy of me which is actually about my stepdad who is not my biological dad. It would not, then, be my genealogy, it'd be my stepdad's, who isn't my dad.