This question, of course, depends on how you define a “priest.” In the culture of His day, Judaism (which was serious affected by both the exilic period as well as roman occupation) recognized only Levites as priests. The priesthood was solely the responsibility and privilege of the Levitical family tribe, but that specific designation was somewhat unclear by Jesus’ day. Following the Maccabean revolt and severe period, a new ruling class had emerged, commonly called the Sadducees. Simon, one of the sons of Judas Macabeus, had become the spiritual advisor as well as the military controller, when his brother was killed in some kind of skirmish. (I may have the first names wrong - I am doing this from memory.) When Simon combined the powers of military control and spiritual leadership, as the next hundred years went by, the influence of the priesthood changed from one of advocacy to one of control. That’s how the high priest could have an army at the time of Jesus.
There is no way possible that Jesus saw Himself as a priest. He did not have access to the Temple hierarchy and process, He did not serve in rotation, He was not in lineage for the priestly functions, and He was from the tribe of Judah, excluding Him permanently from the role. He never spoke of Himself as a priest, nor did He ever perform any priestly duties, according to the understanding at the time of the priesthood.
No evidence within the Gospel record suggests Jesus used priesthood imagery for Himself or His role and identity. His self-disclosure was, to be sure, shocking in its novelty and extreme hyperbole, such as being the son and calling God “Father,” being the Door, the Good Shepherd, the Way, Truth, and Life (allowing no other way to the Father except through Himself) and other such teachings. Never did He use Levitical identities for Himself.
Paul, and later Peter (or possibly earlier, depending on how one dates various writings in the New Testament age) and the writer to the Hebrews, seem to stretch the role of the priesthood to cover all believers as an identity and function, particularly as Christianity moved away from Judaism, were excluded from Judaism, and most of all, after the fall of the Temple of Herod and the destruction of Jerusalem in AD 70. It was, of course, extremely distressing to the Jew of the day that the Temple and the holy city were utterly destroyed, obliterated from the earth, and wiped clean off the mountain. Jerusalem was gone, and gone in such a way that it could not be rebuilt. It was not, of course, until the Moslem inhabitants of Palestine rebuilt a new city based on Arabic values in the 7th century and later. None of what we see in modern Jerusalem is of Jewish origin, except the Wailing Wall which was the foundation of Solomon’s earlier Temple.
So, since the roots and some branches of Christianity were Jewish, the identity and functionality of the levitically-restricted priesthood was stretched into the spiritual work of all believers. Who could object after AD 70? And what did it matter if they did object? No city, no temple, no offerings, no atonement process, no sacrifices, no organized priesthood. The priesthood originally was a metaphoric model for how humans could function with one another under the greater role of The Great and Eternal High Priest Jesus. But I do not think this language is sacerdotal and explicit, but instructive and metaphoric.
