It seems that some people have a distorted notion about the amount of ancient writing which has survived down to our time. They sometimes seem to think that the amount of written material from ancient Jerusalem is comparable to that of a big city today. They seem to imagine historians poring through the stacks of Jerusalem newspapers and police records from April and May, AD 33, and the diaries of Romans and Greeks vacationing in the city...
Newspapers didn't begin to appear until the 17th century, and whatever written records may have been kept by the Roman authorities in 1st-century Jerusalem are gone. We have a handful of such written records of ancient legal proceedings from anywhere in the Roman Empire, mostly from a few sites near the Nile in Egypt. After the actions of the authorities were carried out, the writing involved was thrown away. It doesn't seem to have occurred to people back then to preserve such things. And when papyrus was thrown away, for the most part it rotted away very quickly. Those few sites near the Nile are very dry, which is good for preserving papyrus, and so we have found all sort of written document in garbage dumps, above all the garbage dumps of the Egyptian town Oxyrhynchus. The Dead Sea Scrolls and some other ancient papyri have survived because they were stored in jars.
Most of the ancient Latin writing we have today was written in or fairly near the city of Rome, which was the cultural center of the Empire at the time. But very much even of the writing of the most highly-renowned ancient Roman writers has disappeared over the millennia. The ancient Romans considered Livy their best historian; only about 1/4 of his work has survived. The 2nd-most revered historian in ancient Rome was Tacitus, and 1/2 or more of his work has vanished. And Livy and Tacitus aren't unusual in this regard. This is how much ancient writing has vanished. We have only a fraction of many of the most highly-regarded writers. For many others, we have even less: a sentence or two, or just a mention in someone else's writing, or they've been forgotten altogether. Many of the most highly-regarded ancient writers.
The situation is similar in the case of Athens and the other major cities of ancient Greece. And peoples such as the Jews were much less favored by the Romans than were the Greeks, with the result that more of their culture, including their writing, has disappeared. And the Jews were much better favored than many other ancient peoples, who we only know by their names, or who have been forgotten altogether.
Most ancient Romans didn't know or care much about Judea and Galilee, and in the 1st century, indifference turned to hostility. There are a few lines here and there in ancient Latin and Greek in recognition of the crushing of the Jewish revolt from AD 66-70, and otherwise little mention of the place, except for the work of the authors of the New Testament and a couple of other Jewish writers, Josephus and Philo. And Philo was writing from far away in Alexandria. Without them, the modern world would have completely forgotten about Pontius Pilate until the 20th century, when a stone with a few words about him was excavated in Israel in the 20th century. (And without the New Testament and Josephus and Philo, would anyone today have any idea to whom the stone referred? I'm not asking rhetorically, I don't know the answer.) I keep mentioning the Pilate Stone on this blog because, from the point of view of most Romans of the time, Pilate would have been one of the most important people in Judea or Galilee. And, again, because there is so very little writing which survives from that time and place.
Other than the Pilate Stone and some of the Dead Sea Scrolls, I don't know of ANY writing we have today made in Judea or Galilee during Jesus' lifetime. I would imagine that there are a few more Roman inscriptions, but I don't happen to know. (Words carved into stone are called inscriptions by historians of the ancient Mediterranean world.) There's probably was a lot of writing of various kinds in the Temple in Jerusalem which the Romans destroyed in AD 70. Maybe some more writing will turn up eventually, but for the time being these people who say things like, "We go through all the writings of his contemporaries and there's no mention if him" are talking through their hats: there are no big piles of records to go through. For Jesus' time and place, there are the New Testament and Josephus, and that's pretty much it. Add to that a couple of lines in the works of Tacitus and Suetonius and the younger Pliny, and some of the Dead Sea Scrolls, and the Pilate Stone, and whatever parts of the other papyri found since the late 19th century can be said to have an historical, and not merely an imaginative connection to 1st-century Judea and Galilee. (Remember, most of those papyri have been found in Egypt, near the Nile. Ancient papyrus in most places tends to have rotted away.)