Apart from ignoring Eastern Orthodox interpretations, this is a good application of the FAITH model.
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Many Christians are baffled that Latter-day Saints can study the same Bible they do, and often know the Bible better than they do (Pew Research, 2010), yet hold such a "radically" different theology.
This is the "fish don't know they're in water" principle.
Here's a great case study: "'Mormons' believe in a Great Apostasy. Jesus said in Matthew 16 that His Church would never fall away. They must think Jesus lied!"
Let's unpack that.
When someone looks at the same text I do and comes away with a different conclusion than I did, it is tempting to assume the other person is evil, brainwashed, deceived, or just an idiot.
That is very rarely the case. Instead, it's better to presume the other party is also logical and turn my accusation inward, asking myself:
"What differences in lenses, assumptions, and interpretations do we have that cause us to reach different conclusions? What premises am I taking for granted in my logic?"
Let's start with the text of the verse is referencing-- the words of Jesus to Simon Peter:
"Upon this rock I will build my church; and the gates of hell shall not prevail against it."
His interpretation, which seems understandably obvious to him, is that Jesus promised that there would be no Great Apostasy. Therefore, Mormons are wrong, case closed.
Let's ask, what assumptions does this interpretation rest on that a Latter-day Saint does not share?
1. Biblical Inerrancy
Jesus likely spoke that text in Aramaic. The author remembered those words to the best of his ability when he wrote his Gospel several decades later. His original Gospel was likely written in Hebrew, so even if he could remember the Aramaic, he had to translate it to Hebrew. Then the manuscript was copied over and over again by hand, and eventually translated to Greek by a third party, then copied over and over again.
Matthew's account is the only Gospel to record those words. So the best source we have of Jesus's words is a copy of a copy of a copy of a translation of a copy of a copy of another translation of a copy of a copy of yet another translation of a decades-old memory attributed to (but not definitively from) someone who observed the event.
So we have the first assumption: biblical inerrancy. Many Christians adopt that premise because they think the Bible implies it; Latter-day Saints don't see any such implication, and therefore don't rest their faith on that.
If we found Dead Sea Scrolls part 2 and it had an original Aramaic version of Matthew without that verse, that's fine. Or if it had that verse and it matched exactly, that's fine, too. No sleep loss for me.
2. Interpretation
But let's assume it really was a perfect translation to English. Well, even then we have a problem because English isn't precise, and each person reads it differently based on his or her biases and lenses. We all acknowledge there is some wordplay around "rock" and different-sized stones and Peter's new name. But beyond that, there are several questions the reader needs to answer to understand this verse:
* What is the rock Jesus speaks of?
* What is the church?
* What is the "it" that the gates of hell cannot prevail against?
* What are the gates of hell?
* What does it mean to prevail?
This is where hermeneutics comes in. Different readers with different worldviews will answer these questions in different ways. For example:
The Catholic reading is to interpret the rock as Peter, the church as the authorized institution with priesthood authority, the "it" as that priesthood institution, the gates of hell as all forces aligned against that institution, and prevail as the cessation of that authoritative line.
The Catholic translation, then, is: "Upon you, Peter, I will establish the institution of the priesthood and no evil force will interrupt that institutional priesthood line."
Many Protestants, by contrast, interpret the rock to be the act of confessing Jesus's Lordship as Peter did, the church as the general body of believers (Origen took this view), the "it" meaning Christianity as a whole, the gates of hell as Satan, and prevail meaning a spiritual win.
A Protestant translation, then, is: "Christianity is based on confessions of faith and Satan cannot spiritually win the souls of those who come to such faith."
There are hundreds of variations on this theme:
* Could prevail mean only a permanent victory? Could the gates of Hades/hell have a political/military meaning?
* Could the promise be for the souls' welfare, not for the destiny of the church itself?
* Could the rock maybe even refer to Jesus Himself (as Augustine believed)?
* etc
All good questions. Here is one potential interpretation you might find your Latter-day Saint friends hold:
Jesus is commending Peter for seeking personal revelation from God. That divine, confirming witness is the foundation of a testimony and the bedrock of anyone who would follow Christ (the informal church). The gates of Hades, being a place of horrific pagan sinfulness, may be a symbol of outside forces from which Christians will be safe as long as they seek that personal witness from God.
And persecution from outside forces did not prevail against the institutional Church-- the Church rotted away from the inside due to "itching ears" syndrome.
Or perhaps the church mentioned by Jesus really is the institutional Church and the word "prevail" means only a permanent victory? In which case an 1800-year period is but a small blip on the eternal scale-- certainly within the definition of "quickly" and "soon" the Lord used to describe His Second Coming.
Do you see the complexity of this problem? The Bible is ambiguous—there are dozens of potential interpretations of that verse. Context from surrounding verses, the early manuscripts, etc., all help clarify, but do not definitively answer.
3. Prooftexting
We Christians often treat Bible verses like Pokémon cards-- we assemble our favorites to form a formidable team. And that's good-- it's the scriptures together that help to cut through much of that ambiguity I talked about.
But then we make a little Pokemon army and line them up against the cards our neighbor has.
Remember, if you're going to assemble six verses that (to you) suggest there can't be an apostasy, your opponent can just as easily assemble six verses that (to him) suggest an apostasy wasn't just possible, but predicted in advance. For example:
* Acts 20:29–30
* 2 Thessalonians 2:3
* 1 Timothy 4:1–3
* 2 Timothy 4:3–4
* 2 Peter 2:1–3
* Amos 8:12
* And many more
We could go through the same exercise together of dissecting each verse and battling them, only to find that there are legions of possible interpretations to these verses, and we each bring assumptions and lenses to the text that influence how we choose to interpret it.
That's why, ultimately, Joseph Smith went to the woods to pray. He took the same verse of scripture to a handful of preachers, and none could agree on what it meant. So he went to the One with all the answers. And that's what we invite everyone to do today:
Consider that the verses our critics love to shout at us ad nauseam may not mean exactly what they claim they mean. Consider the lenses you're unknowingly reading the Bible through. Consider the possibility that God has answered much of this ambiguity through more of His word. Study it. Then take it to Him and ask Him directly if it's true.
Or you can keep playing with your Holy Bible Pokémon cards, I guess. But don't overestimate the HP on your Galatians 1:8 and Revelation 22:18 cards-- everyone overplays them stronger than they actually are. 
Responding to:
* On the topic of The Great Apostasy:
If it were true that the Church that Christ built had fallen into apostasy that would mean two things: 1) that Jesus is a liar & 2) if Jesus is a liar and his prophecy failed this means he is not God. Even if he was mortal and just a messenger of God, this would mean his message did not come from God.
What I am referring to is Matthew 16:18 "And I say also unto thee, that thou art Peter, and upon this rock I will build My church; and the gates of hell shall not prevail against it."
You deal with the issue of, if God says the Church will not fall to Hades then why did the Church need to be reformed in the 1830s after centuries of corruption? At what time did the corruption manifest and take hold?
This doesn't mean that corrupt forces cant infiltrate the church, but it does mean it will never fall down into total corruption. It will never teach doctrines contrary to God. If the great apostasy happened like the Mormon church claims, then how can you trust Christ? How can you trust your church? I do not believe the Church ever fell. Anytime there was heresy infesting Christendom the Church would convene at an ecumenical council to stamp out there heresy, this does not mean new teachings were established or new beliefs put in place but rather what has already been revealed is firmly reiterated as dogma for the Church. Many protestants believe the assumption of Mary is a new belief that started by Pope Pius XII in 1950, but this is rather just making a belief that had already existed since the first century firmly set as doctrine to ensure believers can take faith in the truth.
I would agree that people can use religion with the intent of evil and harming others. We see this even in the Bible itself. We see this daily in our world in any branch of faith or religion. However the great apostasy needs to be focused to a specific time. Or at least even a time period within a few decades, and this hasn't been shown at least not in my findings so far.
I'm in complete agreement with the Mormon church that God's church does require an apostolic priesthood with authority bestowed by God and that the protestant reformation lead and will continue to lead many astray. Sadly though I must say the Mormon Church does not meet this requirement of a true apostolic priesthood either. I would be severely immoral if I were to ignore this and not warn you that even if unintentionally, you are being lead astray. If you can not trace a priestly lineage back to the time of the apostles, and can not prove that Jesus instituted a new divine priesthood, then there should be questions raised on the authority of the Mormon church. In my view simply stating that Jesus appeared before Joseph Smith and gave him authority and set up a priesthood is not in itself evidence. In my view, believers of all faiths have a duty to be skeptical and not easily swayed by statements, this is how the Devil can deceive us. If we just simply believe the first bit of information that we either believes holds authority and or makes us feel comfortable we do a disservice to God and to ourselves, we must be analytical and search for evidence. God teaches us to test the spirits and to look at what fruit something bears. I understand why you might be questioning me on my Catholic faith right now and you should. I will address this in comparison a bit later.
I read your post, but I'll focus just on the first paragraph because everything you say after than hinges on that first paragraph being objectively correct... and it's not.
You can't say "Assume A" and then build a large case on that then expect me to agree with your conclusion if I don't agree with the premise. If I agreed with that premise, I'd be a Catholic already
You won’t be able to prove the apostasy occurred or didn’t occur with the Bible alone
Catholics will readily admit they had a string of “greatly apostate” popes
