The problem is simple, the Bible(66 books at least) cannot be interpreted solely by itself in the sens that Protestants usually think of it(Sola Scriptura). Because in reality it is our interpretations which are our final authorities, not the Bible in itself. As a result there are multiple denominations because those interpreting are fallible men looking through their personal, cultural, historical or even mallicious magnifying glasses. Our Lord sort of alluded to this when the Spirit inspired the evangelist to re-tell the history in Mark 9:38-41 :
And John answered him, saying, Master, we saw one casting out devils
in thy name, and he followeth not us: and we forbad him, because he
followeth not us. But Jesus said, Forbid him not: for there is no man
which shall do a miracle in my name, that can lightly speak evil of
me. For he that is not against us is on our part. For whosoever shall
give you a cup of water to drink in my name, because ye belong to
Christ, verily I say unto you, he shall not lose his reward.
Now the solution is also kind of simple as well. And to begin I will not use the Bible as that would be circular. I'll appeal to politics. The one system of government which assures the greatest amount of regularity, of consistency, of unity in politics is the absolute monarchy. Now granted monarchs have done seemingly inconsistent decisions in the past but they are inconsistent from our point of view not theirs. In general it is easier to maintain unity with oneself than it is to do so with multiple other voices, for wherever we are there is also our will.
Now on the Bible, I think we would all agree that Jesus Christ is the King and not the President. That he is advocating dogmatically, ultimately, for a theocracy with him at the top and not something where we get to vote laws in and out as if there was something to perfect.
You must imagine what God, the perfect mind, would set up to get souls saved consistently. ------The most optimal earthly Church structure is what you must imagine. The answer then becomes obvious, because God being all powerful was able to inspire the Bible(at least 66 books) which all three of us understand is possible. It is the most significant mark of unity we have together, that we believe God spoke through men about Jesus Christ the God-man. It is dogma for all three of us, we agree on at least 66 books as dogmatized. Because of inspiration. The key is inspiration. --------------
The key is to have God speak to us consistently and collectively as a group. ---------------------------This then leads us to our Lord's words :
Luke 22: 18-32 (Announcment of a Kingdom(or two ?) at Eucharistic Celebration)
For I say unto you, I will not drink of the fruit of the vine, until the kingdom
of God shall come. 19 And he took bread, and gave thanks, and brake
it, and gave unto them, saying, This is my body which is given for
you: this do in remembrance of me. 20 Likewise also the cup after
supper, saying, This cup is the new testament in my blood, which is
shed for you. 21 But, behold, the hand of him that betrayeth me is
with me on the table. 22 And truly the Son of man goeth, as it was
determined: but woe unto that man by whom he is betrayed! 23 And they
began to enquire among themselves, which of them it was that should do
this thing.
24 And there was also a strife among them, which of them should be accounted the greatest.
25 And he said unto them, The kings of the Gentiles exercise lordship
over them; and they that exercise authority upon them are called
benefactors. 26 But ye shall not be so: but he that is greatest among
you, let him be as the younger; and he that is chief, as he that doth
serve. 27 For whether is greater, he that sitteth at meat, or he that
serveth? is not he that sitteth at meat? but I am among you as he that
serveth.
Ye are they which have continued with me in my temptations.
And I appoint unto you a kingdom, AS my Father hath appointed unto me;
That ye may eat and drink at my table in my kingdom, and sit on thrones judging the twelve tribes of Israel.*
And the Lord said, Simon, Simon, behold, Satan hath desired to have
you(CHRISTIANS), that he may sift you as wheat(CONFUSION): But I have prayed
for thee(Popes), that thy faith fail not: and when thou art converted(IN A STATE OF GRACE),
strengthen thy brethren(CHRISTIANS).
Now please this is important, because usually we get stuck because many of you have difficulties seeing how we could trust in a fallible structure. The point is, we are fallible, the structure is not. Jesus calls the Apostle by the name Simon for a reason. Sometimes Simon was Peter, his supremely authoritative Apostle as he explains when he gives the keys of the Kingdom(again it is a kingdom) after Peter infallibly declares that Jesus is the Son of God(Matthew 16:15-19). But he is also called Satan just a few verses later(Matthew 16:22-23). This exemplifies the bipolar nature of many of our leaders, sometimes they're right sometimes they're wrong. But with Peter it is even more interesting; because God, our God, desired to replicate the structure of the davidic kingdom :
David was born in Bethlehem, as was Jesus.
David started his Kingship at age 30, the same age Jesus started his
ministry on earth.
David, a shepherd, was anointed by Samuel in 1 Samuel 16, and the
Spirit of the Lord descended on David. Jesus, the good shepherd, was
the Messiah, which means “the anointed one,” and the Spirit of God
descended on Jesus during his Baptism in the River Jordan.
Again in 1 Samuel 16, David was responsible for exorcising the evil
spirit from Saul. Jesus also cast out demons.
David was betrayed by one of his trusted advisers, Ahithophel, in 2
Samuel 18, who hanged himself as a result. Jesus was betrayed by one
of his own apostles, Judas, who also hanged himself.
In 2 Samuel 15, David goes up to the Mount of Olives, crying. Jesus
also wept at the Mount of Olives.
David was the King of Israel. Jesus is the King of Kings who sits on
David’s throne(Luke 1:32).
Now, who dealt with business in the abscence of the king? Who was second in command so to say? The davidic kingdom also had a prime
minister who had the absolute authority of the King himself.
In Isaiah 22:15-24 we learn that God is angry with his prime minister
Shebna, and will replace him with Eliakim, who will be given the keys
to the Kingdom, and will have the power to open and shut, who will be
a father figure to his people, and who will be a sure peg on a throne
of honor to his father. The whole weight of his father’s house shall
rest on him. Jesus gave Peter the keys to the Kingdom of God, a symbol
of authority prefigured by Eliakim in the OT. The word "Pope" means
Papa, and Peter and his successors have the power to bind and loose
(open and shut) in the name of Jesus, from Matthew 18:18. The entire
weight of the Catholic Church rests on the Pope and his teachings,
just like the entire weight of David's house rested on Eliakim.
(http://www.catholicbible101.com/thechurchthenewisrael.htm)
To sum up, whatever the Popes have bound on Earth as infallible
concerning faith or morals has ALREADY BEEN bound in Heaven by Jesus
Christ and we simply discovered it in the continum of time by God's
grace. This duality is guaranteed as it clearly makes explicit in the
Bible. This then means that the Church, the body of believers and the
Mystical Body of Christ, is regulated by 3 essential factors all
working together in the incapacity of contradicting one another. --------The
Holy Bible, The Holy Tradition and The Magisterium(teaching office:
Pope + Bishops united to him). This layout of 3 bases forming one
single base is, historically even in Church History, the most stable
of all Church structures. Whereas Protestantism has split and
duplicated itself denying its own fathers sometimes, and Orthodoxy has
tried to maintain unity but is became schismatic even within itself.
Note this : The Church is divided in three parts, the first one is on Earth fighting for the salvation of souls(Church Militant), the second is in Purgation(Church Penitent, their works are being tested(1 Cor 3:15)), the third is in Heaven interceeding for those of us in need(Church Trumphant, Heb 12:23).
Lastly, to conclude, I'd like to point out the structure used by the first Christians by presenting to you the Pentarchy. It is a hierarchical system agreed upon even by the Orthodox and it's very interesting. 
Pentarchy (from the Greek Πενταρχία, pentarchía, from πέντε pénte, "five", and ἄρχειν archein, "to rule") is a model of Church organization historically championed in the Eastern Orthodox Church. It found its fullest expression in the laws of Emperor Justinian I of the Byzantine Empire. In the model, the Christian church is governed by the heads (patriarchs) of the five major episcopal sees of the Roman Empire: Rome, Constantinople, Alexandria, Antioch, and Jerusalem.[2]
[...]
The greater authority of these sees in relation to others was tied to their political and ecclesiastical prominence; all were located in important cities and regions of the Roman Empire and were important centers of the Christian Church. Rome, Alexandria and Antioch were prominent from the time of early Christianity, while Constantinople came to the fore upon becoming the imperial residence in the 4th century. Thereafter it was consistently ranked just after Rome.
Irenaeus
But since it would be too long to enumerate in such a volume as this the succession of all the churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion, assemble other than where it is proper,
by pointing out here the successions of the bishops of the greatest and most ancient church known to all, founded and organized at Rome by the two most glorious apostles. Peter and Paul, that church which has the tradition and the faith which comes down to us after having been announced to men by the apostles.
With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition (Against Heresies 3:3:2 [A.D. 189]).
Cyprian of Carthage
"Cyprian to [Pope] Cornelius, his brother. Greeting. . . . We decided to send and are sending a letter to you from all throughout the province [where I am] so that all our colleagues might give their decided approval and support to you and to your communion, that is, to both the unity and the charity of the Catholic Church" (Letters 48:1, 3 [A.D. 253]).
Firmilian
"[Pope] Stephen ... boasts of the place of his episcopate, and contends that he holds the succession from Peter, on whom the foundations of the Church were laid [Matt. 16: 18]. ... Stephen ... announces that he holds by succession the throne of Peter" (collected in Cyprian’s Letters 74[75]:17 [A.D. 253]).
In seeing how some still doubt, I draw a final appeal to authority which cannot be deflected based on emotional appeal. By the covenant,
all are obligated to submit, and this is seen in the old covenant. You
must submit, it is an absolute decree.
Matthew 23: 1-3
Then spake Jesus to the multitude, and to his disciples, Saying The
scribes and the Pharisees sit in Moses' seat: All therefore
whatsoever they bid you observe, that observe and do; but do not ye
after their works: for they say, and do not.
THE PROBLEM : Sola Scriptura(because of human error), and unwillingness to submit to the appropriate authorities(schismatic obstinance).
THE SOLUTION :Theocracy of the Church Militant, agreeing with the papacy ushers in a whole set of universal doctrines. Optimal security and ideal for dealing with the problems of life(monasticism, prayer life, missions, preaching) rather than multi-tasking as an amateur theologian responsible for crucial and time consumming theological expositions.
[Now this might not please you, but if God is providential in Church History then the debate is basically over, it is the answer, til this day 52% of Christians are Catholic]
[I know many still have questions, I recommend Catholic Answers and Shameless Popery]
