“Nobody comes to me unless the Father called him first”.
Life everlasting is what we are created for at the instant of conception. We are called to become saints and to be in heaven.
It so happens that Baptism is the way in which the Lord was pleased to restore the sanctifying grace in our soul and partake of the divine nature (2 Pt 1:4), and that Holy Communion is the way in which we come to be substantially united with Christ in the manner of mystical spouses.
Ultimately, Christ is eternal life So in Baptism “we no longer live, but Christ lives in us”, and in Holy Communion “since there is one bread, we who are many are one body; for we all partake of the one bread…so in Christ we, though many, form one body, and each member belongs to all the others”. The two Sacraments are not independent but united, to the degree that Christ linked eternal life to them in a very clear way!
We obtain eternal life only from Christian Baptism. If we die before we have received Eucharist (as would be the case with many (Latin Rite) children who die, but it applies to everyone), we are still saved (provided we don’t have any post-Baptismal unconfessed mortal sins holding us back).
Your point is well taken, of course: you mean Baptism is “the gateway to life in the Spirit” (CCC 1213).
However, I think it is not accurate to say “we obtain eternal life only from Christian Baptism”. First, the sanctifying grace infused in Baptism can be lost (as you point out in the mention of post-Baptismal mortal sins). Second, our Lord did state very clearly:
Then Jesus said to them: Amen, amen I say unto you: Except you eat the flesh of the Son of man, and drink his blood, you shall not have life in you.
One can argue that someone who dies without having received the fullness of the Christian initiation (Holy Communion and Confirmation at the hands of the bishop) is not a stranger to the hope of salvation by virtue of his Baptismal grace. However, Holy Communion is a life-giving Sacrament…in fact the Catechism reminds us:
[t]he Eucharist is "the source and summit of the Christian life. The other sacraments, and indeed all ecclesiastical ministries and works of the apostolate, are bound up with the Eucharist and are oriented toward it.
For in the blessed Eucharist is contained the whole spiritual good of the Church, namely Christ himself, our Pasch.
The Eucharist is the efficacious sign and sublime cause of that communion in the divine life and that unity of the People of God by which the Church is kept in being.
[It is t]he Holy Sacrifice, because it makes present the one sacrifice of Christ the Savior and includes the Church’s offering.
And by not judging and by forgiving (I could go on…)
All of them are included because they relate to the pouring of sanctifying grace in our heart! The gate is narrow to Salvation, for Christ taught us so…yet the burden is sweet, and the gifts of faith, hope, love, and final perseverance are just that: gifts!
One could argue quite favorably, because it is the doctrine of the Church:
By Baptism all sins are forgiven, original sin and all personal sins, as well as all punishment for sin. In those who have been reborn nothing remains that would impede their entry into the Kingdom of God, neither Adam’s sin, nor personal sin, nor the consequences of sin, the gravest of which is separation from God. CCC 1263].
The Catechism referenced this doctrine from the Ecumenical Council of Florence:
Also, the souls of those who have incurred no stain of sin whatsoever after baptism, as well as souls who after incurring the stain of sin have been cleansed whether in their bodies or outside their bodies, as was stated above, are straightaway received into heaven and clearly behold the triune God. Session 6]
The Catechism rightly speaks of the importance of Eucharist, but never of its necessity.
So, how does this square with the Bread of Life discourse? I really don’t know. But, as we do know, sacraments have “ordinary” and “extraordinary” (ie, not ordinary) dimensions. For example, the ordinary minister of Baptism is a priest or Bishop (deacons are delegated). But anyone may validly baptize if necessary. So, I would say that Jesus is describing the ordinary means (just as he described the ordinary means of water Baptism to Nicodemus, omitting the extraordinary form (Baptism of Desire), which the Church explained later).