For the same reason it isn’t applied more often to Jesus, the New Testament high priest–because most of these priests, like Jesus, were not from the tribe of Levi. In Jewish circles, the idea of a priest not being from Levi was absurd. Everyone knew God had given the priesthood to Aaron and his descendants (Ex 28:1, cf. Nm 16-17).
Most Christian presbyter-priests were not from Levi, much less the Aaronic line, and it would have posed apologetic difficulties for Christians in Jewish communities to refer to their ministers as “priests.” An ordinary, first-century Jew would snort at that idea, saying, “Oh, yeah. Your ministers are priests. They aren’t even Levites!”
The early Christians faced the same problem when it came to the fact that Jesus is the New Testament high priest. Jesus was from the tribe of Judah, not Levi. A first-century Jew would scoff at that idea, too.
So when the Church was still largely Jewish, the priesthood of Jesus and his ministers was kept in the background, and the Greek word for “priest” was used for them only rarely. In that way non-Christian Jews would not automatically reject Christianity and could become familiar with it before being hit with the idea of non-Levitical priests.
Thus there is only one book–Hebrews–which directly refers to Jesus as a priest and only one book–Romans–which directly refers to his ministers as priests. Other books of the New Testament show Jesus and the presbyters doing jobs only priests can do, but the term hiereus is not used for them.
When Jesus’ priesthood is directly stated, the author must go to great pains to justify the idea to Jews. Non-Christian Jews were arguing that Christianity could not possibly be true because Jesus could not be the high priest of the New Covenant. He was from the wrong tribe: He was not a Levite. To reclaim his Hebrew Christian readers, who were in danger of going back to Judaism, the author of Hebrews had to show that this fact did not matter.
That is the basic function of Hebrews chapter 7. It is okay for Jesus to be a high priest because he was not a priest of the order of Aaron but of the order of Melchizedek (Heb 6:20), an order which was older than the Aaronic one (7:1), which did not require a special genealogy (7:3), which was superior to the Aaronic order (7:4-10), which was prophesied to arise again one day (7:11, cf. Ps. 110:4), and which required “a change in the law as well … For it is evident that our Lord was descended from Judah, and in connection with that tribe Moses said nothing about priests” (7:12-14).
The writer wanted to keep his readers from going back to Judaism, and so he had to prove that “it was fitting that we should have such a high priest, holy, blameless, unstained, separated from sinners, exalted above the heavens. Indeed, the law appoints men in their weakness as high priests, but the word of the oath [Ps. 110:4], which came later than the law, appoints a Son who has been made perfect for ever” (7:26, 28).
Once the Church ceased to be mostly Jewish, this was no longer an apologetic problem. Gentiles did not have the idea that priests had to be from the tribe of Levi, so they could convert without this being an issue. Thus, after the Church became mostly Gentile, the priesthood of Christ and his ministers became more prominent.