If we don't believe the existing sources --
there are no historical events to believe.
All the sources agree (no discrepancy) that Pilate hesitated but then pronounced the death sentence -- so his hesitation is likely part of the original event.
When you say "all sources", what you mean is all the sources that still exist.
That's always the meaning, no matter whether it's about Julius Caesar or Alexander the Great or Plato or Zoroaster or Mohammed.
Any claims that there were other sources also which are now lost can be tested over time, because there are new manuscripts still being discovered, as an ongoing fact of today's research. Until new documents are discovered which contradict the current-known sources, it's reasonable to believe the current sources, on any points where these agree. And there are HUNDREDS (thousands?) of points where the current-known sources agree, and also hundreds where they disagree, including on controversial points which someone in power could have suppressed.
But there's very little evidence of someone in power -- or conspirators -- picking out certain documents, or excerpts in them, and censoring or destroying them in order to cover up something they wanted to hide from public knowledge. The best evidence of anything like that going on 2000 years ago is
hysteria and paranoia among today's debunker fanatics
on a crusade to rewrite history
and make it fit their theories about what should have happened.
No one has any argument why we should not accept these accounts as sources just the same as we accept all other ancient written accounts to determine the events near the time when they were written.
During the centuries before those particular stories were canonized there were plenty of reasons to whitewash the part played by the Romans.
We're still waiting for any evidence that there was such whitewashing. There might be evidence of a pro-Roman element possibly added later, but not of an early anti-Roman element being censored from any written accounts. For this conspiracy theory to be maintained, we need evidence of Jesus making anti-Roman statements, inciting a mob against Rome, organizing a military strike force to lead against a Roman garrison, etc.
Whatever whitewashing there may have been must have happened prior to 100 or 200 AD when the writings were in their final form. You can't rewrite history based only on your paranoia. You must have some facts to confirm your conspiracy theories rather than only the paranoia and conspiracy theory as your premise not to be questioned.
There is plenty of evidence, based on facts -- in writings before 300 AD, inside and outside the NT -- that Jesus was opposed by a group (or groups) of Jews at that time. These facts and evidence confirm the Gospel accounts which say some Jews accused him before Pilate. And nothing about this denies that Pilate's main worry was the threat of insurrection against Rome. It is dogmatic prejudice to insist that there could not have been any Jews present who accused him. There is no reason to insist on this absolutist dogma that only Romans had any complaint against Jesus. If this were the case, then there's no explanation how he was brought before Pilate to be accused. Who brought him there? Did Roman soldiers arrest him? No, the evidence is that it was the Jerusalem Temple police who came for him and made the arrest, obeying their orders from the Temple authorities. So we only know the Jewish authorities acted against him.
It's a virtual certainty that the immediate occasion to accuse him was the earlier riot at the temple, where the accounts say Jesus committed violent criminal acts in assaulting innocent persons and destroying property. It was only the Jewish authorities who responded to this, not the Romans. Whatever the details are, it has to be admitted that there was some conflict between Jesus and the Jewish authorities, and this led to the accusations being brought, rather than some military uprising against Romans. Even if Jesus did instigate that riot, which is debatable, there's no evidence of him organizing an insurrectionist military threat to Rome.
But there's no particular reason to believe that whoever produced the Gospels had detailed accuracy as a goal, or an accurate account of the proceedings.
"accuracy as a goal"?
"as a goal" -- one goal among others? Of course there's reason to believe this was a goal.
Absolutely there's evidence that they wanted accuracy -- just as historians like Josephus and Herodotus and others wanted accuracy and yet created their own dialogues and put words into the mouths of the historical characters. They believed their dialogue composition was an honest representation of what the characters actually said and were thinking when they interacted with each other. Seldom do we know the writers wanted to distort the truth of what happened or was said. Rather, we assume the writers exaggerated what this or that character said, but seldom that they deliberately distorted the facts or misrepresented the character or made up their own historical facts. In some cases they made mistakes, but they did not fabricate "facts" to slander a character and promote their prejudice in contradiction to the real facts.
"Quite the contrary" -- what?
After you say "quite the contrary," you're supposed to tell us what shows that the opposite is the case, i.e., that the Gospel writers did not want any accuracy.
There are examples to show how the Gospel writers sometimes presented their account from their existing sources and stayed true to their source rather than changing it, and as a result, their final version might even be less intelligible in places where their source had missed something so that the story is incomplete. This means the writer/editor/redactor stuck to his source, wanting to present as much as possible but not wanting to fabricate something inaccurate to fill in the gap.
episode from the "trial"
The "trial" scenario gives an example of this, where all three synoptic gospels leave something out, leaving the episode unclear, and yet the whole scene is made clear by taking the pieces from each account and fitting them together so they make sense with all the pieces combined. This example is about the guards harassing Jesus:
Mt ch. 26
67 Then they spat in his face, and struck him; and some slapped him,
68 saying, "Prophesy to us, you Christ! Who is it that struck you?"
Mk ch. 14
65 And some began to spit on him, and to cover his face, and to strike him, saying to him, "Prophesy!" And the guards received him with blows.
Lk ch. 22
63 Now the men who were holding Jesus mocked him and beat him;
64 they also blindfolded him and asked him, "Prophesy! Who is it that struck you?"
Note that in each of the above 3 versions of this scene, taken alone without the others, this episode makes no sense. Each account leaves out something necessary in order for the whole scene to make sense. If you read each one separately, it's not clear what the point is.
In the Mt version, why do they ask him "Who is it that struck you?" When do you smack someone and then ask "Who is it that struck you?" What sense does that make? None. To ask someone you just smacked who it was that smacked them -- when would you do that even if you wanted to smack that person? What are you adding by asking them who struck them?
But notice the Mk version -- they command him to "Prophesy!" What does the covering his face and striking him have to do with prophesying? Prophesy what? What kind of prophesying are they demanding from him? What does striking him while his face is covered have to do with prophesying? Again, taken alone, this version of the incident doesn't make sense. Something is left out.
The Luke version almost makes sense. But here too there's a disconnect, because they put on the blindfold AFTER they already struck him, so that
he could see who it was who struck him at that earlier moment, so he could simply tell them who it was who had struck him before they put the blindfold on him. For the scene to make sense they have to wait until
after the blindfold is on before striking him and then asking him who it was who struck him.
The scene makes sense only if the blindfolding happens first, then second the striking, and third asking him to "prophesy" or identify who struck him. The "prophesy" refers to his (psychic) ability to see through the blindfold to identify which one of them struck him. "Prophesy" doesn't mean only to predict the future, but also to speak something by means of some psychic power, or magic, or divine or supernatural ability, etc.
So, each version left out something necessary: Matthew left out the reference to the blindfold, and Mark left out what the "Prophesy!" means -- ordering Jesus to tell them who struck him, and Luke gets the blindfolding in the wrong order by placing it AFTER they struck him rather than before and so making the "Prophesy!" meaningless.
Since each version of this makes no sense by itself, why did the writer give us his account of it? What did the writer think he was telling the reader about the event? It appears that each writer individually did not understand what was happening, but he thought it was important anyway and should be included,
because it was in his sources. Each Gospel writer here thought it was important to report what he had in his limited version of it and so stay true to his sources, in case the facts do matter, regardless that he didn't see the point of it in the version he presents.
This tells us the Gospel writers did care about the facts, based on what evidence they had, and would diligently give us the best they could, from whatever they had from earlier reports, because the earlier sources, closer to the event itself, are more reliable than the later redactors or editors or revisionists trying to compose a good story for the readers.
This example from the "trial" also tells us that the Gospel writers did not get together and compare notes in order to provide us with a unified story compiled by consensus between them, polished and slickened up for maximum effective impact. The original "Good News" was something which just happened, maybe partly in confusion, without someone following any organized masterplan, and those reporting it both earlier and later all thought it was urgent to get the facts of it out there, reporting it accurately and expeditiously in accord with the best current means of communicating it.
So, even if these accounts we have now are deficient on some points or contain discrepancies or bad interpretations, you cannot deny that they're driven by a concern to retrieve the original facts or details as best as possible, even to the point of including something they didn't understand, writing of it later, but which might be important and traces back to the original events or to the earliest accounts of it, putting maximum priority on the original facts or events to be communicated accurately. And then, conforming to these priorities, each one shaped his interpretation of what happened.
This doesn't mean nothing ever was "made up" by these writers. It means we detect also an impulse to stick to the cold facts, regardless of any theory about what "the story" should be even if the writers also had a "program" of some kind in their head. They thought this was about the earlier historical events, during the term of Pontius Pilate, and about reporting the facts accurately.