Many of us fail to realize that it was never God's intention to have a "Priesthood" per se. It was never His desire to require a human being to stand as a mediator between Him and His people. And such a misunderstanding also contributes to a lot of confusion on the Old and New Covenants.
Before Sinai, God's people apparently came directly to Him. Cain and Abel brought their offerings directly to the Lord (Gen 4:3ff). Noah built an altar and sacrificed to God (Gen 8:20). Abraham, Jacob, and, of course, Moses, built altars to the Lord. All of these individuals seemed to serve as their own priests.
God wanted direct contact with all of His people. He wasn't like the pagan gods, who angrily required a human mediator to stand between him and his people. But, unfortunately, God's people did not understand His character. Having lived in Egyptian bondage for so long, they were convinced that Yahweh was vengeful, exacting, and needed to be appeased. So in the shadows of Sinai, they said to Moses, "You speak with us, and we will hear; but let not God speak with us, lest we die" (Exo 20:19).
Of course, since God is humble, He relented and set up a priesthood - a group of men that would intercede and mediate on behalf of the people. Instead of communicating directly with His people - which He wanted to do all along - He had "go-betweens" that relayed any communication to the Children of Israel.
Since Christ came, we now understand this more clearly. We are all "priests" (1 Peter 2:9) and Jesus is our great High Priest, Himself (Heb 6:20). We do not need another human being to stand as a mediator between God and ourselves, and any system of worship that requires a priest to stand as a mediator between us and God is based on paganism. Such are truly living under the "Old Covenant."
As I said earlier as well, such a misunderstanding of God's original intentions for the priesthood also contribute to our misunderstanding of the Old and New Covenants. The Israelite sanctuary service - with its priesthood, sacrifices, laws, and rituals - is rightly labeled the "Old Covenant," but it was not the result of God's initiative and original plan, but because of the hardness of His people's hearts. Thus, God was not necessarily setting up two systems (or "covenants") of salvation, as many in Christianity would like us to believe. Faith has always been the way by which salvation is appropriated (compare Gen 15:6 and Rom 4:3). God always wanted direct access to His people; He always wanted to write His law directly on the hearts of His people, instead of on tables of stone. It was because of His people's stubbornness that He had to humbly relent to their system.
As always, God mercifully meets us where we are.