Our earliest complete text of the New Testament is from about 200 - 300 years after after the individual documents were first set on paper. We have manuscripts from various dates from then until the invention of printing and even the latest ones seem substantially very similar to the Codex Sinaiticus.
What evidence do you have that the process of copying manuscripts was so much more error prone between 100 and 300 than between 300 and 1000?
But there are almost countless variants in the manuscripts in those later centuries. I think Ehrmann estimates between 200,000 and 400,000 variants, others think it is much higher - Eldon Epp suggests 750,000 and Peter Gurry suggests 500,000 non spelling variants.
So there probably need not be a greater rate of variant for us not to be able to know what the original said with surety.
However to answer your question as to why there may have been a greater rate of variation in the earliest period compared to later, it is not just me who thinks this likely but also mainstream and respected scholars, including Ehrmann. Some reasons.
1. There is typically much greater 'churn' in the earliest drafts of just about any document than in later iterations once the narrative becomes more settled.
2. The earliest copyists were non professionals; later the copying became more professionalised, involving people trained (often Monks) specifically to dot he job).
3. The earliest years are ones where orthodoxy was being
established, while in later centuries orthodoxy was being
maintained. The latter drives preservation of the word, while the former lends to alteration to fit with a developing political/theological position.
4. The nature of the earliest church is that it was effectively homeless - largely nomadic. Accordingly there was no base to check back to. Later the church had a settled centre where major documents would be held, and cold be used as reference to the network of churches and copyists out in the field (so to speak).
5. The early versions were on papyrus which normally lasts just a couple of decades in use (we are very fortunate to have anything left), therefore the trail of earlier version, for reference, is rapidly lost. Later parchment was used which lasts much, much longer so copyists could check back several generations for accuracy which was impossible with papyrus. (see 4 also).
6. The earliest versions would be less likely to have been seen as
sacred documents, prior to formal establishment of church structures, and therefore would have been considered easier to change. Later the absolute and
sacred nature of the works became established meaning that changes (other than simply errors) would have been much more difficult to be altered. You need formal structures to police maintaining a narrative - that was in place later, but not in the earliest decades. In those early days who was going to rip up a new copy deliberately changed by a copyist for political/theological purposes. No-one. The notion of heretical and heresy comes only with orthodoxy and that wasn't established from the get-go.
There you go - just a few examples.