What is the extra value of the body of Christ in the Eucharist when someone is already full of the Holy Spirit?

St. Ignatius of Antioch (c. 110 A.D.)

I have no taste for corruptible food nor for the pleasures of this life. I desire the Bread of God, WHICH IS THE FLESH OF JESUS CHRIST, who was of the seed of David; and for drink I DESIRE HIS BLOOD, which is love incorruptible. (Letter to the Romans 7:3)

Take care, then, to use one Eucharist, so that whatever you do, you do according to God: FOR THERE IS ONE FLESH OF OUR LORD JESUS CHRIST, and one cup IN THE UNION OF HIS BLOOD; one ALTAR, as there is one bishop with the presbytery… (Letter to the Philadelphians 4:1)

They [i.e. the Gnostics] abstain from the Eucharist and from prayer, because they do not confess that THE EUCHARIST IS THE FLESH OF OUR SAVIOR JESUS CHRIST, flesh which suffered for our sins and which the Father, in his goodness, raised up again. (Letter to Smyrnians 7:1)

We can see it was already recognized around the year 100.

Your question: How do the Holy Spirit and the Eucharist relate to each other?

Jesus and the Holy Spirit

12. With his word and with the elements of bread and wine, the Lord himself has given us the essentials of this new worship. The Church, his Bride, is called to celebrate the eucharistic banquet daily in his memory. She thus makes the redeeming sacrifice of her Bridegroom a part of human history and makes it sacramentally present in every culture. This great mystery is celebrated in the liturgical forms which the Church, guided by the Holy Spirit, develops in time and space. (23) We need a renewed awareness of the decisive role played by the Holy Spirit in the evolution of the liturgical form and the deepening understanding of the sacred mysteries. The Paraclete, Christ's first gift to those who believe, (24) already at work in Creation (cf. Gen 1:2), is fully present throughout the life of the incarnate Word: Jesus Christ is conceived by the Virgin Mary by the power of the Holy Spirit (cf. Mt 1:18; Lk 1:35); at the beginning of his public mission, on the banks of the Jordan, he sees the Spirit descend upon him in the form of a dove (cf. Mt 3:16 and parallels); he acts, speaks and rejoices in the Spirit (cf. Lk 10:21), and he can offer himself in the Spirit (cf. Heb 9:14). In the so-called "farewell discourse" reported by John, Jesus clearly relates the gift of his life in the paschal mystery to the gift of the Spirit to his own (cf. Jn 16:7). Once risen, bearing in his flesh the signs of the passion, he can pour out the Spirit upon them (cf. Jn 20:22), making them sharers in his own mission (cf. Jn 20:21). The Spirit would then teach the disciples all things and bring to their remembrance all that Christ had said (cf. Jn 14:26), since it falls to him, as the Spirit of truth (cf. Jn 15:26), to guide the disciples into all truth (cf. Jn 16:13). In the account in Acts, the Spirit descends on the Apostles gathered in prayer with Mary on the day of Pentecost (cf. 2:1-4) and stirs them to undertake the mission of proclaiming the Good News to all peoples. Thus it is through the working of the Spirit that Christ himself continues to be present and active in his Church, starting with her vital centre which is the Eucharist.

The Holy Spirit and the eucharistic celebration

13. Against this backdrop we can understand the decisive role played by the Holy Spirit in the eucharistic celebration, particularly with regard to transubstantiation. An awareness of this is clearly evident in the Fathers of the Church. Saint Cyril of Jerusalem, in his Catecheses, states that we "call upon God in his mercy to send his Holy Spirit upon the offerings before us, to transform the bread into the body of Christ and the wine into the blood of Christ. Whatever the Holy Spirit touches is sanctified and completely transformed" (25). Saint John Chrysostom too notes that the priest invokes the Holy Spirit when he celebrates the sacrifice: (26) like Elijah, the minister calls down the Holy Spirit so that "as grace comes down upon the victim, the souls of all are thereby inflamed" (27). The spiritual life of the faithful can benefit greatly from a better appreciation of the richness of the anaphora: along with the words spoken by Christ at the Last Supper, it contains the epiclesis, the petition to the Father to send down the gift of the Spirit so that the bread and the wine will become the body and blood of Jesus Christ and that "the community as a whole will become ever more the body of Christ" (28). The Spirit invoked by the celebrant upon the gifts of bread and wine placed on the altar is the same Spirit who gathers the faithful "into one body" and makes of them a spiritual offering pleasing to the Father (29).

Reference

I am Catholic, and from personal experience I know that taking the Eucharist in the right conditions gives you an immediate spiritual rehabilitation. You become more Christlike, more loving, more calm, etc.

The Church teaches that it also forgives venial sins. What must be understood is that the Sacrifice of Christ on the cross is the same as the eucharistic sacrifice, the Eucharist re-presents the sacrifice. But it does not kill Jesus over and over again every time it is performed, it's that same sacrifice on the cross which is continually re-presented in churches for the People of God.

CATHECHISM OF THE CATHOLIC CHURCH

Article 3 The Sacrement of the Eucharist

1365 Because it is the memorial of Christ's Passover, the Eucharist is also a sacrifice. The sacrificial character of the Eucharist is manifested in the very words of institution: "This is my body which is given for you" and "This cup which is poured out for you is the New Covenant in my blood."187 In the Eucharist Christ gives us the very body which he gave up for us on the cross, the very blood which he "poured out for many for the forgiveness of sins."188

1366 The Eucharist is thus a sacrifice because it re-presents (makes present) the sacrifice of the cross, because it is its memorial and because it applies its fruit:

[Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper "on the night when he was betrayed," [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.189

Now in the case of someone who's as holy as St. Paul the Eucharist is still necessitated ordinarily.
I say ordinarily because it is possible to get a pass if you are animated by extraordinary grace like many protestants I would think. It should also be noted that ordinary grace is more reliable, more consistent. Of course in this case we will stick to the ordinary grace scenario.

In the Garden of Eden, Adam and Eve were kept alive because of the fruits of the Tree of Life(Gen 3:22). In Revelation, which is the final conclusion and a miror image of Genesis, we learn that the nations are kept alive by the fruits of the Tree of Life(Rev 2:7, Rev 22:2). We know that no mere tree can keep someone alive eternally, but we also know that God likes to use physical things to usher in spiritual realities. Like the Bread and Wine of the Holy Communion.

John 6:48-50 King James Version (KJV)

I am that bread of life. Your fathers did eat manna in the wilderness, and are dead. This is the bread which cometh down from heaven, that a man may eat thereof, and not die.

Taking this data into consideration, we can conclude that any ordinary ministry must be assisted by the Eucharist even if it shows saintly spiritual maturity. The Holy Spirit is not at odds with the Eucharist, but rather cooperates with the Eucharist to "decode", as it were, all the power which is hidden behind the physical apparence of the sacrifice. Imagine the spiritual layout of your body, your body is the temple of the Holy Spirit, when you take the Eucharist you allow the Holy Spirit to fill you more, to take control of areas which were kept prisoners by habitual sins or ever demonic entities like St. Paul mentions(2 Cor 12:7-9). This is why one must not be in mortal sin when taking the Eucharist, it angers God and he may very well stike us dead as he could have done with Adam and Eve but instead sealed the Garden.

That is how the Eucharist became the center of the Catholic faith. The Eucharist re-presents Christ our Lord, and it sustains us spiritually no matter how holy one may be, in the case of ordinary ministry, we must re-fill and that's how God desired it to be. It keeps us humble and closer to him.