As the NT apostles and faithful members of the NT church were being killed by those under the influence of the prince of this world, the rest of the members of the Church were being led away by the false teachings of "wolves in sheep's clothing" ie false Christian teachers. So what is left is just the members following the false Christian teachers. A shell of the NT church, with no apostolic authority, no real spiritual gifts, full of false doctrines, and no real authority from Heaven. By the time Rome tried to organize all the remnants of these Christian factions in the 4th century, the NT Church had long been dead.
Quotes from Christian reformers:
Martin Luther:
I do not say that I am a prophet. I simply say that they will have to be afraid ofthis as long as they scorn me and heed themselves.... If I am not a prophet I am at least sure of this, that the word of God is with me, and not with them, for I have the Scriptures on my side while they have only their own teachings. ... But do I not preach a new doctrine? No. I simply say that Christianity has ceased to exist among those who should have preserved it--the bishops and scholars.
John M. Todd, Martin Luther: A Biographical Study (Newman Press, 1964), p. 188 (ellipses are original), as cited by Kevin Barney, "Did Luther Think He Lived Duriing a Time of Apostasy?," FAIRLDS.org (archived from 2005) and as cited by Gary P. Gillum in "'Jesus, the Very Thought of Thee': Understanding the Christ of the Restoration and the Reformation," in Revelation, Reason, and Faith: Essays in Honor of Truman G. Madsen edited by Daniel C. Peterson, Donald W. Parry, and Stephen D. Ricks (Provo, Utah: FARNS, 2002).
John Wesley also bemoaned the loss of the spiritual gifts (manifestations of the Holy Ghost) in the early church after two or three centuries. He noted that reports of such gifts became especially rare after the Constantine became a Christian. He wrote that the Christians "had only a dead form left" without the original and extraordinary gifts of the Holy Ghost that were found in the original Church (Sermon 89 in The Works of John Wesley, Sermons 71-114, ed. A.C. Outler, Nashville, Abingdon Press, 1986, 3:263-264, as cited by Winwood, pp. 12-13).
Roger Williams, founder of Rhode Island, was an Anglican who later founded the Baptist church in America. Williams believed that divinely-given authority to men to act in the name of God (which we call the priesthood) had been lost from the earth. He looked forward to a time when Christ would send "new apostles to recover and restore all the ordinances and churches of Christ out of the ruins of antichristian apostasy." (See John Catton, A Reply to Mr. Williams ..., ed. J. Lewis Diman, in Complete Writings, 2:14, 50, cited by Donald Skaggs, Roger Williams' Dream for America, Peter Lang Publ., NY, 1993, p. 43; as cited by R. I. Winwood, Take Heed That Ye Be Not Deceived, R.I. Winwood, SLC, UT, 1995, p. 12)
The message of The Church of Jesus Christ of Latter Day Saints is that God has restored the NT church that the above reformers were seeking.