The writer has led his hearers towards a shocking conclusion. Having argued that God's new covenant in Jesus is better than the old covenant in Moses and that his priesthood is better than the Levitical priesthood, another question is begged - "you can't be saying that we don't need human priests anymore?!!" His answer is in Heb.8:3-13.

This is a different type of question and potentially much more disturbing than the previous one. This is where the rubber hits the road because priests and priesthood were intimately woven into the fabric of society. They were people you knew. You had believed (correctly) for hundreds of years that their role in mediating between you and God was essential for being God's people. So any argument that they might now be redundant has huge consequences for society, for organised religion and for the nature of the people's relationship with God. Wow!

The passage works by comparing what a human high priest did with what Jesus has done. The high priest:

mediates the agreement between the people and God (8:6)
so that the people walk in the promises of God on which that covenant is based
by being appointed to offer gifts and sacrifices as prescribed by the Law of Moses (v3)
in a sanctuary which is an Earthly representation of Heaven (v5)

Jesus:

mediates a superior agreement between people and God (8:6)
so that the people walk in better promises on which the new covenant is based
by being appointed to offer a better and final sacrifice (9:12) of his own blood
in a real sanctuary, the real Holy of Holies (9:11)

So, yes, the ministry of human priests - apart from Jesus, who is THE human - is now superfluous. Not because we no longer have priesthood, but because we have the final, perfect High Priest. The writer shows that God promised this in Jeremiah 31 with a lengthy quote about the New Covenant. The main thrust is that God always promised a time when everyone would be taught by the Lord (NB this doesn't mean that there are no teachers in the people of God anymore, it means there are no more mediating teachers. Teachers help us walk with God, but they don't mediate between us and God. We can have a relationship with God without them).

People use the word "priest" today in different kinds of ways. I am staying with Anglican friends right now who call their church leader a priest without any sense of him having a mediatorial role. It is just their word, as it is for many Anglicans. But other people - most notably many Roman Catholics (for whom I have a high regard on many issues, especially social justice) - still believe that it is essential to have a human priest as to mediate between us and God on a daily basis. Hebrews will have none of this. To rely on a mediatorial priest today is to deny the High Priesthood of Jesus and settle for less than New Covenant Christianity.

Let me sum it up with three questions:

Do people continue to need a human priest to mediate between us and God? Answer: yes, and it is Jesus Christ (who is also God)
Do we still have any other human leaders in the church? Answer: yes, God gives gifts of spiritual leadership to churches to build them up
Is this human leadership priesthood, in that it exercises a mediating role between people and God? No, only Jesus does that. We have access to the Father through the Son, and no longer have any human priests in that sense

I am currently writing a book chapter on spiritual leadership. I believe in leadership. I think the New Testament has a clear role for leaders in the Church. But it isn't a mediatorial role. Anyone who says "you have to come through me to get to God" is - tragically, criminally - putting themselves in the place of Jesus Christ.