According to Catholic Church, why did John the Evangelist not mention the Institution of Holy Eucharist?

Peter F. Ellis, a Catholic scholar, made a pretty convincing case in his 1984 book 'The Genius of Saint John: A Composition Critical Commentary of the Fourth Gospel' that the Fourth Gospel does not attempt to present the events of Christ's life in a narrative, chronological order. Instead, he shows that the Evangelist uses a structured chiastic parallelism that was a common literary method at the time it was written. This literary structure draws attention to the theological content the author wishes to emphasize.

He's not the first, but he is the one who showed conclusively that the entire Gospel is not all "out of order due to the material having been shuffled around", as many had suggested for centuries, but is rather precisely arranged in an ingenious literary structure that doesn't even begin to attempt to tell the story of Jesus in a chronological narrative. Rather, it tells the story of Jesus with a literary structure pointing to the greatest theological truths about Jesus.

From the summary on the back of my softcover copy:

The commentary is based upon the discovery that the Gospel is structured and written from beginning to end according to the well-known laws of parallelism used regularly by ancient authors in the writing of books both biblical and profane. That discovery establishes the architectonic structure and unity of the Gospel, illuminates the literary artistry of the author, and solves with remarkable ease many of the hitherto insoluble problems of the Fourth Gospel.

The commentary, which is complete but not compendious, clarifies text after text and episode after episode, relates the text to the situation and purpose of the author, confirms the innate consistency of the evangelist's inspired message, and testifies throughout to the literary and theological genius of the fourth evangelist.

Please look past the second break below, following a rather lengthy section regarding details of the entire literary structure of the Gospel, to see how that structure determines the placement of the material found in John 6:31-63.

More on Ellis' analysis of John's literary structure.

In the case of the Gospel of John, Ellis shows that there is a five-fingered structure to the Gospel, and each of the five parts also have a same substructure of five fingers.

The Prologue (Jn 1:1-18) is the introduction to the Gospel and is not included in the overall structure of five major parts, but it also has a five-fingered structure of its own.

Part I and Part V are parallel (that is, they deal with similar themes, symbols, or words). Parts II and IV are parallel. Part III stands in the center at the theological summit.

Parts I, II, IV, and V are each further divided into five sections (a-b-c-b'-a') : I-a and I-a' have common elements, symbols, or themes, as do I-b and I-b', I-c is nested in the middle of the parallel sections. Parts II, IV, and V are the same. Further, section I-a is not only parallel with I-a', it is even more parallel with V-a'. I-b is parallel with V-b', I-c with V-c, I-b' with V-b, and I-a' with V-a.

Thus there is an A-B-C-B'-A' structure. Each of those five major parts (Parts I - II - III - IV - V) have an a-b-c-b'-a' sub-structure, and those sections have further a-b-c-b'-a' patterns.

Ellis divides the entire Gospel (plus the Prologue, which has a chiastic structure of its own) into 21 sequences. Sequence 1 is Part I-section a in the above structure, Sequence 2 is I-b, Sequence 3 is I-c, Sequence 4 is I-b', Sequence 5 is I-a', Sequence 6 is Part II - section a, Sequence 7 is II-b, and so on...

Sequence 1 is parallel to Sequence 21, Sequence 2 is parallel to 20, 3 to 19, 4 to 18, 5 to 17, 6 to 16, 7/15, 8/14, 9/13, 10/12, and at the heart of the structure is Sequence 11, which is also Part III in its entirety.

Thus, John 1:19-51 (Sequence 1) is about the first coming of Jesus into public view at the Jordan River. John the Baptist and his disciples bear witness. Simon is called Peter. Nathaniel and two other unnamed disciples are present.

John 20:19-21:25 (Sequence 21) is at the Lake. It talks about Jesus' second coming. Thomas witnesses. Peter is told to "feed my sheep". Two unnamed disciples and Nathaniel are present.

John 2:1-12 (Sequence 2): Mary at Cana. "Woman, what have you to do with me?" Nuptial situation. Water to wine.

John 21:1-18 (Sequence 20): Mary at the tomb. "Mary, why are you weeping?" Nuptial language quoted from the Song of Songs.

John 2:13-25 (Sequence 3): "Destroy this temple (Jesus' body)..."

John chapters 18-19 (Sequence 19): Jesus' body is destroyed in the passion.

John 3:1-21 (Sequence 4): Discourse at night to Nicodemus on eternal life, discipleship, water, and the Spirit.

John chapters 13-17 (Sequence 18): Supper discourse at night on washing of feet, discipleship, eternal life, and the Spirit.

John 3:22-4:3 (Sequence 5): The Baptist repeats his witness about Jesus.

John 12:12-50 (Sequence 17): Palm Sunday, the crowd witnesses about Jesus.

And so on...

Notice how sequence 1 is parallel to sequence 21, but also parallel (in a slightly different way) to sequence 5. Sequence 2 is parallel to 20, but also shares the theme of water and beginnings (weddings - births) with Sequence 4.

Part III (Sequence 11) stands in the middle alone. It's the climax of the entire book. John 6:16-21: Jesus walks on the sea; proclaims, "I Am He"; and then brings the New Israel to the other shore of the sea. Thus the entire literary structure of the Gospel of John points decisively to Christ's self proclamation that "I AM!"
(Greek: ἐγώ εἰμι - literally "I I[myself] am"). That's as close as the NT ever gets to a Koine equivalent for God's personal name revealed to Moses at the burning bush: YHWH יהוה.

John 6:22-72 is Sequence 12 and immediately follows the climax of the entire book, Sequence 11. Compare Sequence 10 (6:1-15) and Sequence 12 (6:22-72) which immediately precede and follow the central message.

Seq. 10: Multiplication of the loaves near Passover
Seq. 12: The loaves are explained as the Eucharist near Passover

They are placed in those positions because they are considered by the writer to be the most central things to the core of the Gospel revealed in 6:20 - that Jesus is God Incarnate, and that his true disciples accept him as such. (6:21 "Then they were glad to accept him into the boat.")

The Passover was celebrated to commemorate the Exodus from bondage in Egypt. Like Moses, Jesus goes into the hills alone. Like YHWH, Jesus feeds the people in the wilderness. Like the Hebrews wandering in the Wilderness, like the synagogue leaders in Jerusalem, the Galilean Jews misunderstand, but in a different way. They see Jesus as a prophet. They desire to make him a king, by physical force if necessary. But they do not recognize him for who he really is.

As the Prologue has already said: "He came to his own home, and his own people received him not. But to all who received him, who believe in his name, he gave power to become children of God; who were born not of blood nor of the will of the flesh nor the will of man, but of God."

Then the disciples are in the boat in the middle of a stormy sea that night. They see Jesus coming across the waves to them and they are frightened. He reveals himself to them like he never has before: "I AM. Do not be afraid." They are glad to accept him into the boat. Immediately they are safe on the other side of the sea.

It's so subtle it's almost easy to miss if one hasn't been paying attention to the literary structure of the entire Gospel. That literary structure is pointing bright, flashing neon signs at these six simple verses. The New Exodus is at hand for those who accept Jesus as God Incarnate.

Contrast the disciples' acceptance of Jesus' claim with the members of the synagogue in Jerusalem when he tells them, "Before Abraham was, I AM." They tried to stone him to death. Contrast the disciples assurance that they are safe when "I AM" gets in the boat with them to the soldiers who come to arrest Jesus in the Garden. When they ask which one is Jesus he tells them, "I AM" and they shrink back in fear like they think lightning is about to strike Jesus dead for blaspheming the Holy name.

After this New Exodus John immediately returns to the theme of the Passover. But this time it is about the new Passover. It's again about those who reject the message Jesus has. "How can he have come down from heaven as he claims, when we know this is Joseph and Mary's son who came from Nazareth?" Then it focuses once more on those who do not turn away, but instead believe.

Jesus asks the twelve, "Will you also go away?" Simon Peter answers, "Lord to whom shall we go? You have the words of eternal life; and we have believed, and have come to know, that you are the Holy One of God."

Move another step away from 6:16-21 in each direction.

Seq. 9 (Jn 5:1-47): At Feast, Jesus cures a paralytic and makes himself equal to the Father
Seq. 13 (Jn 7:1-8:58): At Feast, Jesus refers to his cure of the paralytic and makes himself equal to the Father

Chapters 13-17 are part of a later sequence that is parallel to Nicodemus' visit to Jesus.

Seq. 4 (Jn 3:1-21): Discourse at night to Nicodemus on eternal life, discipleship, water, and the Spirit
Seq. 18 (Jn chs 13-17): Supper discourse at night on washing of feet, discipleship, eternal life, and the Spirit