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The doctrine of the priesthood of all believers is commonly accounted a cardinal Reformation position. Thus, I found this quotation from the Venerable Bede (Beda) in his exposition of Revelation 1:6 interesting
Because the King of kings and heavenly Priest united us unto His own body by offering Himself for us, there is not one of the saints who has not spiritually the office of priesthood, in that he is a member of the eternal Priest
memoirandremains 
The Catholic Church also teaches that all believers of Christ are priests, whether they are males or females – they become priests through Sacrament of Baptism. Thus it is not the cardinal position of Reformation as you wrote.
Why does the Church still retain special priesthood (or ministerial priesthood) comprising male only members and specially ordained for that purpose while New Testament does not even mention such priesthood? The answer is in the Old Testament. Universal priesthood of all believers is not something newly introduced in New Testament, but it is already prefigured in Exodus 19:6. Yet God also established special priesthood from the House of Aaron (brother of Moses) of the Levi tribe (Exodus 28:1, 29:1) – they are known as Levitical priesthood. Whenever Scripture mentions priests and levites (2 Chronicles 23:4, Ezra 3:12, Nehemiah 13:30, John 1:19 etc.) it refers to this priesthood (the levites are those of the tribe of Levi but not belonging to House of Aaron).
The Bible prophesied that God will perpetuate this priesthood, i.e. they will continue offering sacrifices (Jeremiah 33:18, 21-22). This prophecy cannot be fulfilled by Levitical priesthood of Judaism, who still exist today. All male Jews with surname Cohen (or Cohn or Coen) were born priests (kohen is Hebrew word for priest). After the destruction of Jerusalem Temple in c. 70 AD., the sacrificial system of Judaism came to an end. The Bible also prophesied in Isaiah 66:21 that Levitical priesthood will be extended to include all nations, i.e. they don’t have to be descendants of Aaron of the tribe Levi.
Thank you for your comment.
I would be interested (and I am not doubting you) in the source for the nature of priesthood within Roman Catholicism. A doctrine could be quite important for both a Reformed and Roman Catholic understanding, but have nuances which distinguish the two positions.
The doctrine was very important in the Reformed and Lutheran understanding, but I imagine it also differs in certain respects from the Roman Catholic position. I would be interested to read more on this point.
You are welcome to visit my blog. I have a post related to this issue. The title is “Sacrifice of the Mass: Catholic Invention or Reformers’ delusion?”