NO! It is not "a strong indicator of their independence", it is poisitive evidence that what they are writing is HEARSAY and NOT first hand knowledge.
No, what no-one has ever said is that the writers weren't eye-witnesses of the events - what they have said is that they weren't the disciples Matthew and John; or Mark, who the early Church understood to be the lad who was following Jesus and the disciples when Jesus was arrested in the Garden of Gethsemane - Mark 14: 51-2
So far there is not a single item of evidence that was written at the time of Christ - it is all recollection - even the four gospels were written after the passage of years and human memory is fallble!
Unfortunately, there is no evience that there is no such evidence, Owl. Yes, I realise that that may sound a tad tortuous, but with the earliest extant document or fragment dating from about 150 years after the death and resurrection events, t is actually very difficult to know when the materials were originally penned. If, as suggested by some, Mark
was the lad who followed Jesus and the disciples to Gethsemane, the fact that the commonly held date for his Gospel - the early 60s AD - is 30-odd years after the events is largely irrelevant, as he would have been an eye-witness. The added suggestion, that he acted a scribe for Peter's recollections also point to the possibility of eye-witness accounts. Human memory is not as fallible as you like to make out when momentous events occur.
Only something written at the time is viable evidence - and there isn't a single item available that qualifies!
In which case, most of the histories of the First and Second World Wars, most major world events of the last 3000 years, etc. are invalid.
Anything written that might contradict the Church's version of history was re-written, hidden in the Vatican archives or destroyed yonks ago!
OK, evidence for these three claims please. Remember, too, that hiding them 'in the Vatican archives' smacks more of wanting to preserve that destroy them.