The Male Priesthood
As I understand the line of thinking from many proponents of the ordination of women to the priesthood, it first assumes a functional understanding of the priesthood, i.e., a role someone does, and then a political understanding of why it is restricted to men, i.e., a power interest. From these assumptions they do not see how a woman can not do what a man does should she have the talents, training and motivation for the vocation, nor why she would be excluded in principle except by unjust discrimination by power-invested hierarchs (or perhaps, for some, from a motive of mindless tradition).
But this thinking is about as far removed from the understanding of the Church as one can get. Why? Because the Church understands the Sacraments to operate by reason of their symbolism. And what they "operate" is a supernatural reality beyond the entire natural order.
Let's start with Baptism: There is nothing in the element of water as to its natural properties or human uses that can have a supernatural effect by itself. It makes natural life possible (a few days without it and we're dead); it cleans bodies, etc.. There is much that it can symbolize for us from these uses, and from the places and movement of it where it is found and from human associations. But those symbolisms have no supernatural meaning or import of themselves. But God has chosen (aptly, of course) to take this element and employ it for supernatural results through a symbolism He defines by words and actions: Baptism (literally, "immersion") in the Name of the Father, Son, and Holy Spirit. This association of moving water (i.e., Living Water) with the Triune Name of God and the action of immersion or pouring or sprinkling symbolizes by reason of a meaning God imposes, immersion into the Paschal Mystery of the Death and Resurrection of Christ, incorporation therefore into Him, becoming a member of the Church by such an incorporation, and most importantly a Divine birth in the soul by grace, such that the Trinity really dwells there, as in a living temple, infusing His own life, and many Divine powers of Faith, Hope Charity, the Gifts of the Holy Spirit, other virtues, etc., removing all sin, Original or personal, and allowing one to receive the other Sacraments.
Again consider the following re Baptism:
(1) None of its effects -- which are all supernatural -- can occur, could occur, except by the express Choice of God.
(2) All of these effects occur by the symbolism which God puts into the action and words of Baptism, a symbolism connected with the actual human being, Jesus Christ.
Let's suppose someone asks for Baptism but water is unavailable, but we have some vegetable oil. The suggestion is to use the oil instead of water. Result? Nothing! Why? Because of (1) and
(2) There are no natural properties of any created element that can produce a supernatural effect; neither can human willing make a symbolism (through intention) so as to produce any supernatural effect. Only God can do that.
Likewise, in the case of the Sacrament of the Eucharist, we're in
Turning to the priesthood: God chose twelve men to symbolize Christ as High Priest (vis-a-vis both God and the Church) as Head of the Body and as Groom to the Bride. (See especially Ephesians 5 toward the end of the chapter on the Great Mystery and spousal symbolism.) Women can not symbolize the Head (the Man Jesus Christ) to the Body, nor the Groom (the same Man Jesus Christ) to the Bride (the Church).
I wish to point out that not even any man can symbolize this. Only those chosen by the Church and ordained through apostolic succession. So going back to the example of the Eucharist, should a community of Catholics lack an ordained man (it happened in both Japan and Korea for hundreds of years, as well as in much of England and Scandinavia during the same period), they could not have a layman, or ordained deacon even, celebrate the Mass with any supernatural effect. Reason: supernatural causes alone bring supernatural effects and God has chosen to bring about the supernatural effect of extending the Priesthood of His Incarnate Son only through the symbolism He choose for Christ as Head to Body and Groom to Bride through the laying on of hands upon men by the Twelve and their successors.
Just as it is not by the natural properties of water or the human symbolism of Baptism (the Jews had a baptism of repentance) that the supernatural effects of Sacramental Baptism occur, neither is it by the natural qualifications of any human being, male or female, as such that any supernatural effects of Priesthood take place. That is why it is not by the natural qualifications of a woman functioning in the role of a priest (preaching, ministering, etc.) that there would by any supernatural effects. Only the men the Church accepts as living symbols of the God-Man Jesus Christ as Head to the Body of the Church, and Groom to the Church as His Bride through sacramental Ordination by Apostolic Succession have the supernatural causality.
It is obvious that a woman can not symbolize being Christ the Groom (inescapably male); neither also the supernatural Headship (i.e., as reference point and absolute origin -- rather than superiority - like the Father to the Son and Holy Spirit in the Trinity). So it is not a put-down of the natural or attained abilities of a woman who could function in the activities of the office as well - or in some cases (preaching) better -- than a man or particular men from a social point of view. It is the lack of symbolism in being a woman of the specific Man Jesus Christ Who as a man is Priest, Head to the Body, Groom to the Bride, Son to the Father, Father of the World to Come. The woman can of course symbolize the Body and the Bride of Christ and she does, and Mary is the Woman and the symbol of the Church in its highest.
That is why Christ chose only men to attend the Last Supper where He established the Eucharist and Priesthood. To the objection that "only Jews were there, so only Jews should be priests" the answer is easy: the Divine symbolism of Christ as Head and Groom is not removed by the absence of Jewishness, while it is by the absence of a) a Baptized male believer of sufficient age, b) being chosen and ordained by Apostolic Succession, through the laying on of hands for the office. And that is how the Church took it from the beginning with the ordination of gentile men while the Apostles were still alive and refraining from having priestesses.
I might point out that this tradition of the office of the priesthood as a Sacrament, a sacred and supernatural symbol of Christ, i.e., the specific man Jesus Who is God Incarnate, is why the faithful have put up with great deficiencies in their priests down through the ages, because they sense the supernatural effects are through the symbolism and not through the natural or attained qualities of the individual priest. The Sacrifice of the Mass is offered, the Body and Blood of Christ is transubstantiated, sins are forgiven, by a validly ordained priest through the Divinely chosen symbolic actions of the Sacraments, regardless if the priest himself is a jerk, or worse. That human condition doesn't stop the supernatural effects from taking place between Christ and the faithful. That is also, I may add, why Bishops sometimes value even bad priests in desperate circumstances, if through discreteness no scandal or moral harm takes place (although America seems to have reached new lows in a very few cases recently).
So the Church does not see itself competent to ordain women. Neither does it see itself competent to allow deacons or laymen to officiate at the Eucharist or forgive sins, or replace water for wine in the Eucharist, or oil for water in Baptism. There would be no supernatural effects if she did make those changes since it is not by the functioning of natural or attained talents or human intentions, but by the Divine chosen symbolism that Divine Mysteries are mediated in the Church. Thus it is not a put down or injustice of women to discriminate on the basis of supernatural symbolism received by the Church from God Himself. And the natural or attained talents of women can be expressed in other ways to great effect as they have been from the beginning.
And I might add that this sacramental awareness of the Priesthood also corrects the political understanding of Church leadership as primarily a position of power. It is, rather, a position of authority: kingship, Priesthood (the sanctifying role of offering the sacrifice) and prophecy (teaching with God's authority God's Mind on things, all as service to the flock, not as a political power in society. Protestants deny all this. The minister is chosen solely on the basis of a community's evaluation of his talents and functions towards their needs; they go to him for that and remove him when they don't see the results they want. In that understanding of ministry it really would be male chauvinism to insist on excluding women on principle or because that's the way it always was done in the past (traditionalism rather than Tradition).