Judaism has a priesthood. It is hereditary, it descends in the male line from Aaron, and the rabbi does not belong to it by virtue of being a rabbi.
The confusion is understandable, because the rabbi occupies the position that a visitor expects a priest to occupy: he stands at the front, he speaks, he is the person you ask. But nothing he does requires him. Ten adult Jews constitute a minyan and may pray. Any competent member may lead the service, read from the Torah, or blow the shofar. A wedding requires a valid contract and witnesses. A funeral requires a burial. None of these require a rabbi, and in communities without one they proceed.
What the rabbi has is semikhah, a certification that he is competent to rule on questions of Jewish law. His authority is juridical and scholarly. When he is needed, it is because a decision has to be made: is this permitted, is this valid, is this document correctly written.
The kohen, meanwhile, is still a kohen. He is called first to the Torah. He pronounces the priestly blessing. He avoids corpse impurity, which is why some Jewish men do not enter cemeteries. He may not marry a divorcee. He does none of this because anyone appointed him and nobody can take it away, because it was never given.
The offices did not merge. One survived the destruction of the Temple with its functions intact but suspended, and the other was built to replace what the Temple had done.