That is not a fact at all. The Temple never contained all the genealogical records of the Jews, ever. Furthermore, if that were the case and only such records could be used, then that would eliminate Jesus as the "Messiah" too, as follows: The First Temple was destroyed during the Babylonian conquest. This happened hundreds of years before Jesus' time. If only Temple genealogical records applied then Jesus' genealogical pedigree would also be worthless. In addition Jesus would be eliminated because one or the other, or both, of the genealogical records given in the gospels would be at variance from your putative Temple records. Therefore one of the gospels that claimed to present his genealogy would be in error. But the gospels are claimed to be inerrant. The discrepancy would nullify the claims of the gospels and destroy the basis of Christianity.Among the Jews, the family tree is very important, which can be seen just from the fact that the OT contains many family trees. For Jews family trees are identification, they show who a person is, to whom he belongs. Family trees are also proof for heirs, because in order to inherit something from an ancestor, the descendant must prove that he is the rightful heir and this is proven by a family tree. And that is the point, for we know that the Messiah is the rightful heir of King David and thus ascends the throne of David. Now the fact is that all the genealogical records of the Jews were destroyed by the Romans during the destruction of the temple. Only one family tree survived, the family tree of Jesus in the NT which goes from David to him. With this family tree one can prove that Jesus is a descendant of David and therefore has the right to the kingship of David. He is the only one who can do this, all other Jews living today have no family tree that leads up to David. If a Jew shows up and claims to be the Messiah but is not Jesus, he cannot prove that he is the descendant of David because he has no family tree.
Ask yourself, why did God destroy all the genealogies of the Jews except that of Jesus. He did it so that the world would know that there can be no other Messiah except Jesus.
You might find the following interesting: Jewish genealogy-Outreach Judaism