It's important to understand how the documents of the New Testament would have originated and initially circulated. We shouldn't think that an author would compose a document on his own, without telling anybody else much or anything about it, then send that one copy away and pay little attention to what happened with it afterward. It's not as though Paul would write a document like 1 Corinthians without anybody else knowing of it before its completion, then would send that one copy out and do little or nothing to track its status thereafter. If such a scenario had occurred, then we could imagine somebody getting hold of that one copy and significantly altering it without anybody else's knowledge. It would still be highly unlikely that such corruption would occur with all or even most of the New Testament under such conditions, but theories of widespread textual corruption would at least be more reasonable under those conditions.
In reality, document production in antiquity was often a highly public procedure. And some of the New Testament documents explicitly reflect that fact. To use 1 Corinthians as an example again, Paul has a co-author (1:1), other people with him (16:19-20), and a scribe (16:21). Commenting on the gospel of John, Craig Keener notes:
"Besides any skills John had acquired [which could change at different times in his life], he undoubtedly would have had help; even the most literate normally used scribes, and Josephus’s staff included style editors to improve his Greek. John would have been an unusual writer if he published the work entirely by himself." (The Gospel Of John: A Commentary, Vol. 1 [Peabody, Massachusetts: Hendrickson Publishers, 2003], pp. 101-102)
It was common for documents to be read and commented upon publicly prior to their wider circulation. Steve Mason comments:
"An author normally composed a work gradually and by constant revision, presenting it in stages to ever-widening concentric circles, moving from closest friends to more remote associates through a combination of oral recitation and distribution of partial drafts. The cycle of oral presentations typically began in the intimate setting of a private residence, perhaps at a dinner party, and moved as the author gained confidence in the work to rented auditoriums. The oral dimensions of this entire process, even with written texts, should always be kept in mind." (Josephus, Judea, And Christian Origins [Peabody, Massachusetts: Hendrickson Publishers, 2009], p. 52)
And authors commonly kept a copy of their document before sending out another copy to be circulated:
"There is also evidence that even private letters regularly had copies made (e.g., Cicero, Familial Letters 9.26.1; 7.18.1; Letters to Atticus 13.6.3)....As noted above, scholars for a number of years have suggested that Paul might have made copies of his letters at the time he was writing them with his scribe and missionary companions. This would follow the pattern of many ancient writers - among them, Seneca and Cicero as literary authors (who speak of actual letters, not composites made out of the fragments of earlier letters), and Zenon as a documentary writer - who made copies of their letters before having them dispatched. This allowed the writers not only to refer to their letters in the future - perhaps explaining why 1 and 2 Thessalonians, and Colossians and Ephesians, among others, have verbal material in common - but to have the copies either with them or in the possession of their companions....Paul is widely regarded in classical studies as one of the great letter writers of the ancient world. If that is true - and his corpus of letters argues that it is - then it is logical to think that Paul followed the conventions of ancient letter writing, including producing copies." (Stanley Porter, in Craig Evans and Emanuel Tov, edd., Exploring The Origins Of The Bible [Grand Rapids, Michigan: Baker Academic, 2008], pp. 189-190, 195, n. 106 on p. 195)
Thus, an author didn't entirely give up control of the transmission of his text to other people. He kept a copy himself and could restart the copying process anytime he wanted with his own edition of the original.
Once a community received a document like a letter from Paul, the document would commonly be read publicly (1 Thessalonians 5:27). Thus, even those who were illiterate could become witnesses to the original text by means of hearing it read publicly early on.
Authors often took steps to ensure the preservation of their text and to monitor the status of the text's circulation. Thus, ancient authors often commented on subjects like what titles were being applied to their works in libraries, how some people were interpreting their work inaccurately, how some people were altering their text, etc. Their concern over the text didn't end once the first copy was sent out. Thus, the second-century Christian Dionysius of Corinth wrote:
"For I wrote letters when the brethren requested me to write. And these letters the apostles of the devil have filled with tares, taking away some things and adding others, for whom a woe is in store. It is not wonderful, then, if some have attempted to adulterate the Lord’s writings, when they have formed designs against those which are not such." (cited in Eusebius, Church History, 4:23)
"I adjure thee who mayest copy this book, by our Lord Jesus Christ, and by his glorious advent when he comes to judge the living and the dead, to compare what thou shalt write, and correct it carefully by this manuscript, and also to write this adjuration, and place it in the copy." (cited in Eusebius, Church History, 5:20)
Such concern by authors for the preservation of their work was common. The idea that men like Dionysius and Irenaeus would have had such concern and would have monitored the circulation of their work, yet men of far higher authority and influence in the church (such as the apostles) didn't, is absurd.
The apostles and other authors and relevant witnesses would have lived into the late first century and beyond. See, for example, Richard Bauckham's Jesus And The Eyewitnesses (Grand Rapids, Michigan: Eerdmans, 2006).