A similar statement of "everyone" or every true Jew being a priest is present in the notion of ἔλεγχος = conviction "being the real priest" entering the hearts of all Jews:
And like this is the injunction given respecting the house in which it happens that leprosy often arises; for Moses says that, "If there be a taint of leprosy in the house, the owner shall come, and shall tell it to the priest, saying there is something like a taint of leprosy has been seen by me in my house,"34 and presently he adds, "And the priest shall command him to dismantle his house, before the priest enters into the house to see it, and all the things that are in the house shall not be impure; and after that the priest shall enter the house to examine it." (132) Therefore, before the priest enters in, the things in the house are pure; but after he has entered in, from that time forth they are all impure. And yet the contrary would have been natural, that when a man thoroughly purified and perfect, who is in the habit of offering up prayers and purifications, and sacrifices for all the people come into a house, all that is therein would be improved by his presence, and would become pure from having been impure; but now they do not even remain in the same condition as before, but they are brought into a worse state by the arrival of the priest. (133) But whether this is consistent with the literal and obvious order of the words, those men may inquire who are in the habit of, and fond of pursuing such investigations; but we must affirm distinctly, that no one thing can be more consistent with another than the fact, that when the priest enters in, all the things in the house should be polluted; (134) for as long as the divine word has not come to our souls as to a hearth of hospitality, all its actions are blameless; for the overseer, or the father, or the teacher, or whatever else it may be fit to entitle the priest, by whom alone it is possible for it to be admonished and chastened, is at a distance: and those persons are to be pardoned who do wrong from inexperience, out of ignorance what they ought to do: for they do not look upon their deeds in the light of sins, but even sometimes they believe that they are doing right in cases in which they are erring greatly; (135) but when the real priest, conviction, enters our hearts, like a most pure ray of light, then we think that the designs which we have cherished within our souls are not pure, and we see that our actions are liable to blame, and deserving of reproach, though we did them through ignorance of what was right. All these things, therefore, the priest, that is to say, conviction (ἔλεγχος), pollutes, and orders that they should be taken away and stripped off, in order that he may see the abode of the soul pure, 35 and, if there are any diseases in it, that he may heal them. (On the Unchangeableness of God)
This is not the only place where Philo identifies as a priestly figure:
It was also greatly honoured by Moses, a man much attached to excellence of all sorts, who described its beauty on the most holy pillars of the law, and wrote it in the hearts of all those who were subject to him, commanding them at the end of each period of six days to keep the seventh holy; abstaining from all other works which are done in the seeking after and providing the means of life, devoting that day to the single object of philosophizing with a view to the improvement of their morals, and the examination of their consciences (καὶ τὸν τοῦ συνειδότος ἔλεγχον): for conscience being seated in the soul as a judge, is not afraid to reprove men, sometimes employing pretty vehement threats; at other times by milder admonitions, using threats in regard to matters where men appear to be disobedient, of deliberate purpose, and admonitions when their offences seem involuntary, through want of foresight, in order to prevent their hereafter offending in a similar manner. (Allegorical 1)
So the purpose of conversion to Judaism is to let ἔλεγχος enter one's heart through some sort of mystical initiation.
And the Lord God called Adam, and said unto him, where art Thou?"{30}{#ge 3:9.} Why now is Adam, alone called, when his wife also was concealed together with him? In the first place we must say that the mind is summoned, and asked where it is. When it is converted, and reproved for its offence (ὅταν ἔλεγχον λαμβάνῃ καὶ ἐπίστασιν τῆς τροπῆς), not only is it summoned itself but all its faculties are also summoned, for without its faculties the mind by itself is found to be naked, and to be absolutely nothing, and one of its faculties is also the outward sense, that is to say the woman. (50) The woman therefore, that is the outward sense is also summoned together with Adam, that is the mind, but separately God does not summon her. Why not? Because being destitute of reason she is incapable of being convicted by herself διὰ τί; ὅτι ἄλογος οὖσα ἔλεγχον ἐξ ἑαυτῆς λαμβάνειν οὐ δύναται). For neither can sight, nor hearing, nor any one of the other external senses be taught, and moreover none of them are capable of receiving the comprehension of things; for the Creator has not made them capable of distinguishing anything but bodies only. But the mind is able to receive teaching: on account of which fact God calls that, but not the external senses.
Clearly we are starting to see that Philo understands that it is not ἔλεγχος per se that is transforming ordinary people into gods or angels but the reception of the divine Logos. In other words, anyone who has the divine Logos is a priest.