I am amazed on your ignorance of Nehemiah writings ...No, they are official copies. And only for the priests. That has nothing to do with establishing pedigree of the Davidic moshiach. Stop trying to put words into Josephus' mouth.
As you note during the Babylonian exile there was no Temple and no Temple copies of priestly genealogical records. Yet Ezra was able to determine who was and wasn't a priest. That's because there were records besides the copies in the Temple. The fact that Ezra was able to correctly determine who was or was not a priest without using Temple records shatters your argument. Indeed Ezra used Jewish genealogical records from other sources to recreate the copies for the Second Temple.
Jewish genealogical records are recorded in multiple locations. One is in scripture, of course, for genealogies up to the time a scripture was written. There are also genealogies records in the Talmuds. Also individual families were commanded to maintain records. This is often done in a Sefer Yuchsin. Jewish genealogical sources are among the most (if not the single most) complete of any people.
Neh. 7:61 And these went up from Tel-meʹlah, Tel-harʹsha, Cheʹrub, Adʹdon, and Imʹmer, but they were unable to verify their paternal house and their origin, as to whether they were Israelites: 62 the sons of De·laʹiah, the sons of To·biʹah, the sons of Ne·koʹda, 642. 63 And of the priests: the sons of Ha·baiʹah, the sons of Hakʹkoz, the sons of Bar·zilʹlai, who took a wife from the daughters of Bar·zilʹlai the Gilʹe·ad·ite and was called by their name. 64 These looked for their records to establish their genealogy, but they could not be found, so they were disqualified from the priesthood. 65 The governor told them that they should not eat from the most holy things until there was a priest who could consult the Uʹrim and Thumʹmim.