How can you know this? How can you know that any 'fine detail' isn't a later addition? How can you know what is or isn't an exaggeration or lie?
The downside of having no real provenance is that you really don't know very much at all.
From Peter Williams' lecture entitled, "New evidence the gospels were based on eyewitness accounts" here is a list of the types of detail which, he argues, the gospels "get right":
Place names
Peoples' names
Plants
Shape of the houses
Shape of the temple
Coinage
Social stratification
Religious setting.
This indicates that the sources were not physically or temporally distant from the events.
Aside from his material, I would add that the later gospels, Luke and Mark, do add material to the earlier one, Matthew. Also, Matthew has itself been edited. But when looking at these additions, we can see that extra detail is sometimes authentic and sometimes not. In Mark, for example, the author often expands a text. For example, in the healing of the paralytic, Mark says that the man was carried by four men, whereas Matthew and Luke don't give that detail. There isn't any way of knowing whether Mark has assumed that there were four, one for each corner of the bed he was on, or whether he got the detail from an eyewitness. Either way it doesn't necessarily affect the truth of the story, he could just have been glossing it up for his readers.
The detail in the list above however is verifiably correct, so I am sure that although some of the detail in the gospels is a later addition, the conclusion that Peter Williams makes still seems justified.
An example he gives, which I shared a few days ago, is how, in the feeding of the 5000, some incidental details in John and Luke that would not convey any meaning if taken in isolation, do in fact explain each other when one is read in the light of the other. This would indicate that it isn't an exaggeration or lie, but was what the people telling the story saw. I'm referring to the disciples making the people sit down in groups of 50, which practically speaking enabled them to be fed, but also it tells us how they knew there were 5000. Also how the location of the incident, stated by Luke, explains Jesus' question to Philip, who was from that town, recorded in John's account.