The Holy Mass as a True and Proper Sacrifice
When dealing with the Eucharistic sacrifice, Mediator Dei reiterates the teaching of the Council of Trent that the Holy Mass is a proper and true sacrifice and not just a memorial of the Passion or the Last Supper:
Christ the Lord, ‘Eternal Priest according to the order of Melchisedech,’ (Ps 109:4) ‘loving His own who were of the world’ (Jn 13:1), ‘at the last supper, on the night He was betrayed, wishing to leave His beloved Spouse, the Church, a visible sacrifice such as the nature of men requires, that would re-present the bloody sacrifice offered once on the cross, and perpetuate its memory to the end of time, and whose salutary virtue might be applied in remitting those sins which we daily commit, …offered His body and blood under the species of bread and wine to God the Father, and under the same species allowed the apostles, whom he at that time constituted the priests of the New Testament, to partake thereof; commanding them and their successors in the priesthood to make the same offering (Council of Trent, 22, 1). (no. 67)
The august sacrifice of the altar, then, is no mere empty commemoration of the passion and death of Jesus Christ, but a true and proper act of sacrifice, whereby the High Priest by an unbloody immolation offers Himself a most acceptable victim to the Eternal Father, as He did upon the cross. ‘It is one and the same victim; the same person now offers it by the ministry of His priests, who then offered Himself on the cross, the manner of offering alone being different (Council of Trent, 22:2).’ (no. 68)
The reason for the latter is that, given the present glorious state of the human nature of Jesus Christ, the shedding of blood is now impossible, so the sacrifice of Christ is manifested outwardly by the separation of the Eucharistic species under which He is present, and which symbolize the bloody separation of His Body and the Blood. “Thus the commemorative representation of His death, which actually took place on Calvary, is repeated in every sacrifice of the altar, seeing that Jesus Christ is symbolically shown by separate symbols to be in a state of victimhood” (no. 70).
Reformers Shift Emphasis to the “Memorial”
This traditional presentation was not to the taste of the innovators, who began to put the accent on the commemoration (although without the nuda commemoratio connotation of the Protestant reformers). Rather, they gave it the meaning of an objective and real memorial that “re-presents” what happened historically and communicates it here and now effectively.
From this new perspective, R. Gerardi explains, “the memorial [celebration] expresses the reality of the event, the ‘objective updating’ and presence of what is commemorated. It is not that it repeats itself, since the event was set historically once for all (ephápax); but it is present. The act of Christ makes its effect felt here and now, committing those who remember it. The sacrifice of Christ was historically performed only once: the Eucharist is his memorial (in the fullest sense of the word), a living presence of grace.”
The aforementioned Jesuit Fr. Martín-Moreno explains why it is not a question of a multiple reiteration of the unique sacrifice of Christ:
It is not that the time of salvation repeats itself here and now, but rather that man here and now enters again and again into communication with a permanent presence that is beyond elapsed time. … In the liturgy, the point of intersection of time and eternity is reached. There, the participant becomes a contemporary of biblical events. Man becomes a contemporary witness of what happened then. Christ is born at Christmas, [and] rises at Easter. Is anamnesis man’s work or God’s? Man is the one who commemorates, but as a human act, his action of remembering cannot transcend time; it cannot enter the time tunnel to return to the past. The divine action alone, transcending time, brings the mysteries to our here and now. That is why the liturgy, before being an action of man, is the action of God (p. 46).
The path had been opened by the pioneering theses of the then Fr. Charles Journet (later created cardinal by Paul VI) and the French philosopher Jacques Maritain, for whom the real presence of Jesus Christ would double as a kind of real presence of His sacrifice.
This theological option in favor of the memorial, which omits that the Mass is a bloodless renewal of the sacrifice of Calvary and affirms that the latter only becomes present during the celebration, offers a weak interpretation of the dogma of faith proclaimed by the Council of Trent. According to this dogma, each Mass is “a proper and true sacrifice” performed in sacramental form because transubstantiation causes the Divine Victim’s Body and Blood to be truly present and symbolically separated. 
Francis Opts for an Extreme Memorializing Concept
Desiderio Desideravi clearly and insistently adopts this theological option of the Mass as a memorial that has the sacrificial aspect only secondarily to the extent that it is a commemoration. Already at the beginning, describing the Last Supper the Lord wanted to eat with the Apostles, Francis says:
He knows that he is the Lamb of that Passover meal; he knows that he is the Passover. This is the absolute newness, the absolute originality, of that Supper, the only truly new thing in history, which renders that Supper unique and for this reason ‘the Last Supper,’ unrepeatable. Nonetheless, his infinite desire to re-establish that communion with us that was and remains his original design, will not be satisfied until every man and woman, from every tribe, tongue, people and nation (Ap 5:9), shall have eaten his Body and drunk his Blood. And for this reason that same Supper will be made present in the celebration of the Eucharist until he returns again (n° 4).
Incidentally, note that, in the document’s first paragraph describing the Mass, in addition to the theory of the one unrepeatable action, the pope affirms that the Mass is a representation of the Supper and not of the sacrifice on Calvary per se. This is reminiscent of the original (defective and subsequently changed) Protestant-leaning definition of the Mass given in the General Instruction on the Roman Missal, to which Ottaviani and Bacci objected so forcefully in their Short Critical Study. It is also worthy of note that this paragraph suggests that every man and woman should or shall eat of the Eucharist, which suggests a soteriological universalism and fits in with Pope Francis’s pragmatic support of any and all Christians—Catholic or not, in a state of grace or not, living by the Decalogue or not—receiving the Eucharist.
Returning to the main theme: Desiderio Desideravi has some references to the sacrifice of Jesus on the cross, but at no point does it say that sacrifice is renewed in a bloodless way at each Mass. On the contrary, while one of the first paragraphs affirms that “the content of the bread broken is the cross of Jesus, his sacrifice of obedience out of love for the Father,” it goes on to say that, after participating in the Last Supper’s ritual anticipation of his death, the Apostles “could have understood what He meant by ‘body offered’, ‘blood poured out’. It is this of which we make memorial in every Eucharist” (n° 7). That would have been the most appropriate time to teach that the Mass is not only a memorial but an unbloody renewal of the sacrifice of Calvary, sacramentally represented in the separation of the Eucharistic species. Pope Francis chose to omit that truth of faith and refer only to the memorial.
A few paragraphs later, the document insists that the Liturgy is not a memorial of the Apostles’ remembrances, but a true encounter with the Risen One (an idea repeated 9 times throughout the document). He continues: “The Liturgy guarantees for us the possibility of such an encounter. For us a vague memory of the Last Supper would do no good. We need to be present at that Supper, to be able to hear his voice, to eat his Body and to drink his Blood. We need Him. In the Eucharist and in all the sacraments we are guaranteed the possibility of encountering the Lord Jesus and of having the power of his Paschal Mystery reach us. The salvific power of the sacrifice of Jesus, his every word, his every gesture, glance, and feeling reaches us through the celebration of the sacraments” (n° 11). Again, note that the emphasis is placed on participation in the Supper and not on being spiritually united to Jesus who offers himself to the Father in sacrifice at each Mass—an aspect entirely left out.
Mass as Remembering
the “Gift” Jesus Gave at the Last Supper?
When speaking of the correct understanding of the dynamism of the Liturgy, Francis uses words already quoted in the previous section, which make it clear that, for him, the sacrificial character of the Mass results from the commemoration of the Passover of Jesus: “The action of the celebration is the place in which, by means of memorial, the Paschal Mystery is made present so that the baptized, through their participation, can experience it in their own lives” (n° 49).
Later, this idea becomes more explicit when referring to the central nucleus of the Mass:
In the Eucharistic prayer—in which also all of the baptized participate by listening with reverence and in silence and intervening with the acclamations (Institutio Generalis Missalis Romani, nn. 78-79)—the one presiding has the strength, in the name of the whole holy people, to remember before the Father the offering of his Son in the Last Supper, so that that immense gift might be rendered newly present on the altar (n° 60).
The text not only completely omits Christ’s offering during the Passion (of which the Supper was a ritual anticipation) and avoids saying that the sacrifice is renewed, but even leaves out the very word “sacrifice” by calling it an “immense gift.”
Add to all of the above the fact that nowhere in Desiderio Desideravi are found expressions such as “transubstantiation,” “real presence,” or analogous formulations indicating that “the Eucharistic Food contains, as all are aware, ‘truly, really and substantially the Body and Blood together with soul and divinity of our Lord Jesus Christ,’” as Pius XII says in his encyclical (No. 129), citing the Council of Trent (sess. 13, can. l.). Nor does it contain anything resembling Mediator Dei’s exhortation that pastors should not allow the faithful to neglect “the adoration of the august Sacrament and visits to our Lord in the tabernacles” and should not allow “churches [to] be closed during the hours not appointed for public functions”—a viewpoint defended by some “who are deceived under the pretext of restoring the liturgy or who idly claim that only liturgical rites are of any real value and dignity” (n° 176).
These unilateral actions are responsible for the disastrous loss (or at least the serious dilution) of faith in the real presence of Our Lord Jesus Christ under the Eucharistic species, confirmed by opinion polls in several countries. The most expressive is by the Pew Research Center, which found that “just one-third of U.S. Catholics agree with their church that Eucharist is the body, blood of Christ.”
 Philippe-Marie Margelidon O.P., en «La théologie du sacrifice eucharistique chez Jacques Maritain», in Revue Thomiste (Enero-Marzo 2015), 101-147.
 Ver Claude Barthe, La Messe de Vatican II—dossier historique (Via Romana, Versailles, 2018), 181.