There is no blanket permission for all Christians to go around administering this Sacrament. James is clear that the "presbyteros ecclesiae" should be called, not just anyone.
As for whether a presbuteros is a priest, I don't know of any other kind of man who can forgive sins: "and if he be in sins, they shall be forgiven him". Likewise, since the Eucharist is a propitiatory sacrifice for sin (cf. Lk 22:19-20; 1 Cor. 10:16-21), it follows that the ministers of the Eucharist are true priests (cf. Heb. 5:1). Ancient and apostolic tradition restricts the celebration of the Eucharist to presbyters and bishops, therefore it follows that they are true priests.
And now as to prescribing who ought to receive, and who to administer this sacrament, this also was not obscurely delivered in the words above cited. For it is there also shown, that the proper ministers of this sacrament are the Presbyters of the Church; by which name are to be understood, in that place, not the elders by age, or the foremost in dignity amongst the people, but, either bishops, or priests by bishops rightly ordained by the imposition of the hands of the priesthood. ... If any one saith, that the Presbyters of the Church, whom blessed James exhorts to be brought to anoint the sick, are not the priests who have been ordained by a bishop, but the elders in each community, and that for this cause a priest alone is not the proper minister of Extreme Unction; let him be anathema. (Council of Trent, Session XIV, Decree and Canons On Extreme Unction)
Jer. 33:17 - Jeremiah prophesies that David shall never lack a man to sit on the throne of the earthly House of Israel. Either this is a false prophecy, or David has a successor of representatives throughout history.
Dan. 2:44 - Daniel prophesies an earthly kingdom that will never be destroyed. Either this is a false prophecy, or the earthly kingdom requires succession.
Isa. 22:20 - in the old Davidic kingdom, Eliakim succeeds Shebna as the chief steward of the household of God. The kingdom employs a mechanism of dynastic succession. King David was dead for centuries, but his kingdom is preserved through a succession of representatives.
Isa. 22:19 - Shebna is described as having an "office" and a "station." An office, in order for it to be an office, has successors. In order for an earthly kingdom to last, a succession of representatives is required. This was the case in the Old Covenant kingdom, and it is the case in the New Covenant kingdom which fulfills the Old Covenant. Jesus our King is in heaven, but He has appointed a chief steward over His household with a plan for a succession of representatives.
Isa. 22:21 - Eliakim is called “father” or “papa” of God's people. The word Pope used by Catholics to describe the chief steward of the earthly kingdom simply means papa or father in Italian. This is why Catholics call the leader of the Church "Pope." The Pope is the father of God's people, the chief steward of the earthly kingdom and Christ's representative on earth.
Isa. 22:22 - we see that the keys of the kingdom pass from Shebna to Eliakim. Thus, the keys are used not only as a symbol of authority, but also to facilitate succession. The keys of Christ's kingdom have passed from Peter to Linus all the way to our current Pope with an unbroken lineage for almost 2,000 years.
Acts 1:20 - we see in the early Church that successors are immediately chosen for the apostles' offices. Just as the Church replaced Judas, it also replaced Peter with a successor after Peter's death.
John 21:15-17; Luke 22:31-32 - Jesus' creation of Peter's office as chief shepherd with the keys passed to Linus, Cletus, Clement I, all the way to our current Holy Father.
Matt. 23:2 - this shows that the Jews understood the importance of succession to the chair and its attendant authority. Here, Jesus respects Moses' seat ("cathedra") of authority which was preserved by succession. In the Church, Peter's seat is called the "cathedra," and when Peter's successor speaks officially on a matter of faith or morals, it may rise to the level of an "ex cathedra" (from the chair) teaching.
Eph. 3:21 - this divine word tells us that Jesus Christ's Church will exist in all generations. Only the Catholic Church can prove by succession such existence.