That's essentially what the gospels were - the story and beliefs of Jesus passed on in the oral tradition of the rabbis before being written down a few generations later
As for like, bureaucratic forms? It was 2000 years ago, so by the time we started to care we basically are left with only whatever happened to be preserved in a collapsed building no one cared to demolish or rebuild - libraries and record halls tend to get burned down over the years. This is at a time when writing was expensive and a rare skill - it would be extremely strange for a record of a trial of a revolutionary run by a Pontius (basically the lowest rank of administrator sent to back water provinces) to have kept detailed records of executions (the Romans were extremely hierarchical and did a lot of executions)
Plus, the movement grew big enough to catch the attention of the local ruler (and the collaborating religious leadership who pushed for his execution) in the span of months. There was every incentive for uprising to be suppressed - it would be an embarrassment that they'd have every incentive to keep quiet
By the time anyone even started to consider that this Jesus guy was more than a run of the mill revolutionary in some backwater the empire barely cared about, it was because the ideas had spread to the point they started to threaten Roman rule. Probably through the Roman legions, who were largely conscripts sent to the other side of the empire "earning" the right to be Roman (part of the reason why there were so many uprisings)
During the time he was alive, no one took up arms or disrupted trade. By the time the nobility even heard his name, it was decades later - and at this point, we do have the odd surviving correspondence mentioning the issue
Frankly, I would be extremely skeptical of any document describing Jesus when he was alive - I think the only record there was a Pontius Pilates is some military discharge record of someone with that name in the right time and with enough honors to corroborate his existence