Two words of caution here. First, the high priesthood of Jesus is based on the ideal forms found in the Bible, not in the high priesthood as it actually functioned in the first century. By way of analogy, we could study the office of president of the United States as it is described in the constitution, or by the way various presidents have functioned as president over the more than two centuries. James Buchanan, for example, usually is ranked at the bottom of the list of presidents by historians, mostly for his handling of the issues which erupted into the Civil War.
We would not, therefore, want to describe the office of president using Buchanan as our example! In the same way, the high priests who held office in the first century were politically motivated and not particularly good examples of the way a priest ought to behave in his office. What is remarkable is that the book of Hebrews does not condemn the current High Priest as corrupt, nor does he say anything negative about the worship of the Temple other than it has been completed in Jesus.
Secondly, the word “priest” has connotations in English which are not present in the function of a Jewish priest. We are not describing a Catholic or Orthodox priest, but rather the Jewish priest. This modern sense of the word is not particularly helpful in understanding the priesthood in the Hebrew Bible. The priest in the Jewish Temple was the mediator between God and man. As such, the office of priest foreshadowed the ministry of Jesus who was provides access to the throne of God for those who have entered into new life through him.
The original intention of a priest in the Hebrew Bible was to be an intermediary between God and Man. The High Priest chosen to enter the Holy of Holies on the Day of Atonement. He represented the nation and did his duty on that Day on behalf of the nation. If the High Priest performed his function right, then the sin of the nation was covered. Since he was a fallible human, there was always the possibility that the atonement was imperfect.
Not so with Jesus as the ultimate High Priest. He is the perfect intermediary between God and man because (Hebrews argues) he was true human – untainted by sin. Therefore he preformed his duties in the real sanctuary properly, providing real atonement to the whole world, once and for all!