This is from my perspective and studies. I have not only studied from LDS sources but from Non LDS sources. I do not want to disrespect, and if I do I apologize first off.
The apostacy did not just occur after the death of the last apostle. It was taking place during the ministry of the apostles. When they would go to a city and convert jews to the gospel, they set up a organization there. But many times because of communications they were left to themselves with only the old law. There was not bible or New testiment to study and glean truths from. This is why Paul wrote letters of correction to the different cities. But you can imagine that Paul could not address all the concerns, and could not correct fast enough. So jewish traditions and outside influences crept in.
When the apostles were killed off faster than they could replace them, the apostacy sped up. For a couple of hunderd years after the death of the last apostle, there was still not a combined book of the New Testament. You had parts of the written gospel here and there. I am not saying that those who in apostacy were evil doers, but that they were doing their best with what they had. The gosple message was so good, and so strong that even though I believe that it had started to be distorted and truths had been lost or replace that it still brought people together. It united people. It caused them to be better than they were. But because I believe that authority to act in Gods name left with the last of the apostles, the whole earth tumbled into the dark ages. For centuries little or no progress was made. Men in power desires were not to improve the people as a whole but only about them gaining more power and control. Under this setting came Constantine. A ruler with a problem of how to unite the people he had control over. He could see that Christianity was a great unifying force. And thus we have religion taking over the control of the people by this. Because of this, wars were started in the name of Christ. There was such a control over the people that if there were differing views about religion, they were either banned or killed. This was a great motivator follow oners leaders. This is a very brief overview, and I am sure that you have heard this many times. This is my view.
I will address one of the views about why I believe Catholics believe there is a unbroken link. I will go to the first prophet of the church after the death of Christ. Peter. The scriptures teach that Christ was asking his disciples who men thought he was. Some said that he was Elijah, or an Elias, or some said he was John the Baptist. But Christ said who do you think it is I am… Peter stated that,“Thou art the Christ, the Son of the Living God”. Christ said, "Blessed art thou, Simon Bar Jonah, For flesh and blood hath not revealed this unto to thee, but my Father which is in Heaven.
He is my understanding. Peter did not know that Jesus was the Christ because Jesus told him so. At that time Jesus was flesh and blood. And as much as someone can tell another truth, as someone here has said, they really do not know 100% that it is true. But Jesus said that through personal revelation from the Father that Peter knew that Jesus was the Christ.
Jesus then went on to say that "And I say also unto thee, That thou art Peter, and upon this rock I will build my church" In my view Christ was not refering to Peter as the rock. But the rock that gave Peter the understanding that Jesus is the Christ. Christ would not build his church on the arm of the flesh, because we know not long after this Peter denied the very Christ and Savior of the world. Not once, but three times. Christ stated that the very gates of Hell would not prevail against it. Again my understanding is that the Gates of Hell are the gates that hold the dead in. Once the gospel has been accepted, and the resurrection has taken place, the gates that hold the dead will be open and make it possible that a reuniting of the spirits and bodies takes place.
That is my understanding of this.