The Eucharist is the most important thing. All the culture, laws, institutions, and offices of the Church are to provide the sacraments, and protect their nature, preserve them intact so that they can be given to the people.
The Blessed Sacrament is the most important and the center of all the other sacraments. It's the worship of the Incarnate Christ, here on earth, and the center of the Gospel.
In the Gospel, Christ gives us baptism, which makes divine life in us possible. Confirmation is the fruition of baptism.
Baptism cleans the house so that Jesus in the Blessed Sacrament can live in us. The Blessed Sacrament is the point of baptism. "Unless you eat the flesh of the Son of Man, and drink His blood, you have no life in you."
So: I'd say if you are baptised, you are baptised into the Church, you are made ready for Divine Life. You are made receptive to the Divine Life of Holy Communion. If you remain oriented towards Our Lord, you retain your baptismal grace. If you sin, you require absolution in the sacrament of Penance. This returns you to the proper orientation. So the orientation to, the preparedness for Jesus is the essence of baptism. But the purpose of that orientation, that preparation is Jesus in the Blessed Sacrament, in the full presence of Him, body, blood, soul and divinity.
Just as when Our Lord exorcised the young man and said that if you drive out seven demons, but do not replace them with anything, with Life, evil spirits with the Holy Spirit, demons will return worse than before. Supernature abhors a vacuum. Just so, we are baptised so that God can make a home in us. And God is present in the flesh in the Blessed Sacrament.
So, even though Baptism brings us into the Church, the fullness of communion is to be at one with the Body of Christ, the Church, by receiving the literal Body of Christ in the Mass. To be in communion with Our Lord, you have to bring Him into the home he has prepared through Baptism. He is brought into you in Holy Communion.
In order to have the Blessed Sacrament, of course, our Eucharistic Liturgy must be apostolic, and we know it is apostolic through the succession of the apostles.
So the office of Apostle, made present in the successors to the Apostles, the bishops, exists to serve the Eucharist. It exists to preserve and keep present to those who need it (namely everyone), the nature of the sacraments, above all the Blessed Sacrament. The world cannot be saved without the sacraments, so their nature must be safeguarded by the apostolic office. The bishop exists to protect and provide the Eucharist.
So, since we need baptism in order to receive the Eucharist, we get baptised. And because we need the apostles to preserve and safeguard the Eucharist, we make sure to stay in communion with the apostolic office, namely the bishop.
Now, the Roman bishop has always had the supreme office of safeguarding the sacraments. As all bishops exist to safeguard the sacraments (the culture of which is the liturgy and the faith/doctrine---"the law of praying is the law of believing"), the Roman bishop as the successor to Peter has always held a unique office, to keep present the work of Peter as head and anchor of the Apostles. As Peter "strengthened the brethren" and "fed my sheep" and was "the rock upon which I will build my Church", he not only maintained and preserved and made available the sacraments, but maintained and preserved and spread the apostolic brotherhood, the order of the apostles and their successors. His final See was Rome, and his successors always, in all the ancient Christian teaching and correspondence from the beginning, were believed by the other bishops to keep present in the world the work of St. Peter: maintaining the unity of the bishops, and in that the unity of their flocks by preserving the nature of the sacraments (and in that the liturgy and doctrine) so that means of salvation would never be extinguished (faith is the means of salvation, but saving faith is always incarnate, that is what a sacrament is: incarnate faith).
Bishops exist to preserve pure and provide readily the means of salvation, the sacraments. The Roman Bishop exists to preserve pure and intact the episcopal office. The bishops protect and provide the sacraments. The Roman Pontiff protects and provides the purity of the episcopal office. Without the bishop, no sacraments. Without the Roman Pontiff, no episcopal purity, no unity, ultimately no episcopacy.
Without Peter, only a vaporized church of warring, pharisaical clerics, as history has always shown. Clerics who always immediately abandon the sacraments (definition of clericalism: sacerdos without sacer, a bishop or priest who neglects sacraments, becomes a cleric, a clerk, a philosopher, a desk-jockey, a podium-jockey, substituting conference tables for sacred altars).
John was a mystic, sure, but why was Peter made the Pope? Because he was the elder brother, he was the father of the Apostles, he was deepest penitent, he was the confessor who could not hold back both his confession of sins, and his confession of faith. John was the first of those "strengthened" by Peter, to see this
powerful, able leader of men, confess his faith and later his love for Jesus. Peter was the pope because he knew within himself that the confession of faith was the confession of sins, the confession that he who says "arise, walk, your sins are forgiven you" is the King, not he who judges, but He who heals. That one must not just confess one's inability to live the law ("the law is a curse") but that one must confess that failure with still the love of Job, Moses, David, blessing the name of the Lawgiver, and confessing one's faith that He will, has healed you. Diagnosis is the work of the priest, not judgement. Diagnosis always followed by medicine, so that he does not "lay on burdens he does not help to carry", the medicine of his own penance, humiliation, and martyrdom, because the medicine the bishop can give is his blood poured out, Christ's blood vicarious.
The faith of St. Peter is to know one's sins, and to confess Sin, one's own sins, and also to confess faith that love for Christ triumphs, that Christ is God and has the power and authority to love you and forgive you. And it is a realization that we can never make up for the evils we have committed against others, and can never take it back, and even if Christ forgives, those whom we have hurt may not. And it is a realization that the only way we can atone for these sins is to forgive those who sin against us, and to repay them with a life of penance for them, and to love our neighbor not as we love ourselves, but as Christ loves him. So the faith of Peter is the determination to "be bound by others and led where you do not want to go", to be crucified with and by his brothers, and as his Lord was, on Vatican hill. To be crucified by a sinner to save his soul and his own together.
It is the humility to ask Jesus to bring him before his brother, John, who saw him run away, and to take up the responsibility to lead, without pride or confidence in his own competence. That is why Peter asks Our Lord on the beach about John. Because John did not run away from Calvary, like Peter did. He is humiliated before John. And that is why he is pope.
So does one have to submit to the governance of the successor of Peter in order to be saved? It seems clear that the Christian faith, the faith of the Gospels, tells us that to be saved one needs the sacraments. That is, one needs the faith of Peter "you are the Christ, the Son of the Living God." Moreover, the faith of the one who confesses not just his own sinfulness, but confesses the power of Jesus to forgive what he has done. Confesses the power and will of Christ that the brotherhood be strengthened and fed. That they may all be one. That the Advocate is more powerful than the Adversary.
To be saved, you must have the faith that the Roman Pontiff offers you, the confession of Peter that the pope keeps present in the world through his office. Yes, absolutely. You must believe in the Incarnate Son of God, and love Him and keep fidelity to Him as far as you are instructed. You must pursue Him, like Peter, leaping naked into the sea and swimming ashore to confess your sins, confess your belief in His power to forgive you, to forgive your enemies, to make you love your enemies, and to heal you, and then eat the food He has prepared on the fire ( of the Holy Ghost), His flesh, His Sacred Heart, Himself, His very friendship, His Person.
Does that mean you must be ecclesially governed by the Roman Pontiff? It seems to me that if you arrive at the conclusion of the logic, but "go away sad" for "this is a hard saying, who can bear it?" then there is a problem. The New Law must arise in our hearts with joy and be taken up as a joyful cross "for my yoke is easy." It is not meant to be imposed. But it is a cross.
You must share the faith of the Roman bishop. I don't know if you have to be governed by him to be saved. But it does seem that when we see that his office was instituted by Christ to maintain the unity of His family, to safeguard the nature of the means of salvation, so that the means by which we are saved, namely the sacraments, will never be snuffed out, then it seems necessary, in the sense that it seems natural, that we must be ruled by the Roman bishop, too.
I don't know if we have to be ruled by him in order to be saved. It does seem that we mustn't rebel against him, knowing what his office is.
The Eucharist is the center of the universal Christian faith. Everything serves that. And since the Roman bishop is the always-and-everywhere protector and provider of the sacraments, it seems natural to be in communion with him not only in spirit but in living as well.
In Scripture, all waters are manifestations of, and forebears and children of the primordial and apocalyptic waters of Baptism. The chaotic abysses of Genesis reappear in the Red Sea, and again rebel against God in the Sea of Galilee. Christ subdues them there through the faith of Peter. Peter alone walks across the chaotic, diseased waters, allowing his faith to place the waters under the Redeemer's dominion. The faith of Peter transforms fallen waves into the salt and water of Baptism. He submerges but is raised up. To be baptised, you are by necessity with Peter because there is no one else out there on the waves but you, Christ, and Peter.
To be saved you must have the faith and confession of St. Peter. It seems only just to have Peter also.