Rabbinic Judaism has preserved the institution of the Aaronic priesthood. However, after AD 70 when the Temple fell,
The Hebrew people already had genealogies because God’s covenant with Abram gave the land of Israel to his descendants Gen 15:18. The genealogies became more important when the children of Israel established tribes Gen 49:28.
They became still more important when God established the Aaronic priesthood. From the beginning, God had required that only Aaron and his descendants could be priests. “And you shall anoint Aaron and his sons, and consecrate them, that they may serve me as priests” Ex 30:30. “And you shall appoint Aaron and his sons, and they shall attend to their priesthood; but if any one else comes near, he shall be put to death” Num 3:10.
Somehow the rigorous genealogies survived the destruction of Solomon’s Temple, so the Jews were able to resume the Temple sacrifices after King Cyrus rebuilt the Temple. God had promised King David that he would send a Mashiakh in his line who would be a Son of God 1 Chron 17:11–14. In Rabbi Yeshua‘s time the rigorous records were still available to prove that he was in the line of David as God had promised Mt 1:1–17; Lk 3:23–38.
However, the genealogies had been preserved in the Second Temple. When it was destroyed all of the great genealogies were burned. Without the Temple there was no sacrificing for the surviving priests to do so the Aaronic priesthood no longer needed ancestry records. The Jewish priests retained mainly honorary roles within Jewish worship and still have them today. For these honorary roles if a man’s name was Cohen (kohen), or if he had a family history as an Aaronic priest, the rabbis accepted him as a presumptive Aaronic priest.
In our time, now that Israel has been restored to the Jewish people, a small number of Jews dream of restoring the ancient Temple, together with its priestly sacrifices required by the Torah. There are a number of difficulties involved, for instance that the Temple Mount is not currently available for large scale construction. However, the greatest difficulty is that the proven line of Aaronic priests ended in AD 70. For Temple sacrifices, some of these Jews believe that a proven line of Aaronic priests would again be necessary, while others would be content with a presumptive priesthood.
Catholics do not anticipate that the ancient Temple will be rebuilt in our time. We hold that Rabbi Yeshua is the new and eternal Temple Jn 2:19–21, with his Final Sacrifice re-presented every day worldwide in the Holy Sacrifice of the Mass until the end of time.