Many radical theories about Christian origins excuse the lack of primary evidence for their reconstruction of events by postulating a successful exercise in suppression of the relevant documents by the early Church. The usual suspect is Constantine, but many Jesus Mythers have realised that to be credible, the suppression must have taken place earlier. The reason for this is the lack of any rebuttals against the Jesus Myth heresy by Christian apologists of the second and third centuries, despite their detailed refutation of literally dozens of other heresies. Also, the many Christian documents to have been found in Egyptian archaeological digs gives us a considerable amount of material uneffected by any edicts of Constantine and his successors.
I think the lack of rebuttals by the early heresiologists is, itself, fatal to the Jesus myth hypothesis. After all, this was a heresy that, according to mythers, was true and so surely was more worth refutation than the weird fantasies of the gnostics. But let me also add a few notes on how successful, or otherwise, the Church has been in suppressing documents it doesn't like.
The medieval papacy has become the very byword for successful control of ideas. The inquisitors were accorded extreme powers to hunt down and destroy subversive literature. There was nowhere to flee, nowhere outside the control of the papacy where Latin literature could realisitically be studied. Almost all literate people were clergy under direct ecclesiatical control. Furthermore, copying manuscripts required skills that hardly existed outside monestaries and universities, not to mention a great deal of money. Surely, in this case, a text specifically condemned and ordered to be burnt by the pope had no chance of survival.
Of course, you already know what I'm going to say. Oddly, condemned documents seem to have a better than usual chance of surviving. Here are two examples: Peter Abelard's Ethica and Theologica were condemned by Innocent II in 1140, worthy only to be burnt. We have an eyewitness account of the bonfire in Rome. Of course, both survive in multiple manuscript copies. Admittedly, there are few from the 12th century, but these multiply in the 13th and 14th. All the power of the medieval church could not prevent its own staff from copying these forbidden works.
The second case is Cecco D'Ascoli, burnt at the stake in 1327. At the same time two of his books, De sphera and Acerba, were thrown to the flames and utterly condemned by the Florentine inquisitor. Of course, they both survive. The rarest of his works is the one that wasn't condemned!
It seems clear that if the medieval church could not stamp out a text of which it disapproved, it is absurd to suggest that the early church was in a position to do so. It is almost as absurd to believe Constantine, in a world with a far higher literary level than the 12th century, could have managed it either. And, in both cases, they leave us no trace at all, even in rebuttals, of the works they allegedly covered up.
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