By "well after the fact", it's still within the lifetime of eyewitnesses. Contrasted with other historical records, it's pretty good. Like Alexander the Great being written about 800 years after the fact, or some details about Julius Caesar being written down 200 years after the fact which nobody disputes. For something we can archaeologically prove which also happened at the time - the pompeii disaster - there is one record 30 years later. Despite it being an event witnessed by hundreds of thousands and likely having influential romans among it's victims. You're really overestimating the frequency of writings and documentation from the first century. In which the New Testament is abnormal in that it has a high frequency. So something that clearly was a big deal did happen. The traditions as well carried across societies, so must have been rooted in fact. As for the larger corpus - those were the centuries later forgeries that were removed for that reason - because they were much later and not seen as reliable. Some of them were attributed to more important figures also, like Thomas. So the early Church clearly cared about accuracy.