This is actually a very interesting topic and I was just reading about the connections between ancient Judaism and Catholicism recently. I think you are bringing up a lot of good points.
I remember learning somewhere that the Church is supposed to be like the Davidic Kingdom of ancient Israel. Christ is the king and the Pope is like the chamberlain so-to-speak if I am remembering correctly. When the Israelite king would leave on some important task, he’d leave the administration of the kingdom in the hands of the chamberlain which he would symbolize by giving the keys of the kingdom to the chamberlain, much like Christ giving the keys of the Kingdom to St. Peter. I think they even referred to the chamberlain as a “father” much like the title of “Holy Father” for the Pope. Another interesting thing was the role of the king’s mother in ancient Israel. Special honor started being given to the “queen mother” starting with Bathsheba and Solomon I think. If you couldn’t get the king to approve sometimes it was better to petition the king through his mother’s intercession. The parallels to the Blessed Virgin Mary should be obvious. These connections are found in either the first or second Book of Kings, although I confess I am not as well versed in Sacred Scripture as I should be ;).
You might be interested in reading a book called “The Crucified Rabbi: Judaism and the Origins of Catholic Christianity” by Taylor Marshall. There were a lot of other parallels to ancient Israel concerning things like the priesthood (as you mentioned), priestly vestments, parishes vs. synagogues, and even parallels to the Eucharist and the Bread of the Presence in the tabernacle in the Jewish Temple (more on this specifically in Jesus and Jewish Roots of the Eucharist by Brant Pitre).
Anyway, I hope you find some of this information useful.