I don't think you understand the nature of scholarship in antiquity. Unlike today, histories weren't composed during or immediately after (in the modern sense) they took place. People's living memories of events preceded the need for such a thing, and writing of any sort, but particularly scholarly histories, were extraordinary undertakings back then.
So you would expect a history of that time period to be published when it was. If it was written significantly earlier, it would actually call into question the credibility of the work.
Tacitus wrote disparagingly of Christians and Jesus, so the idea that he would shill for them by only republishing sectarian documents defies credulity. Further, he is known to have consulted Roman archives for the basis of his work.
But more than that, there is the plethora of writings circulating from very early. We have epistles dated to within 20 years of Jesus' death, and numerous polemical writings that follow during that first century. But in none of the polemical writings is the argument ever made that Jesus didn't exist. So what that would mean is during the time when people were alive that would have witnessed the events in question, the debate was over the nature of Jesus' teachings and the behavior of his followers--never that he didn't exist. Given the heated nature of these debates, it defies logic that the strongest argument that could be used to delegitimize the opposition (that the object of worship by Christians never even existed) was neglected by every polemical writer in antiquity.
Couple that with cross-referencing data mentioned in these early documents to other writings and annals, and the weight of evidence is clear. Then there are developmental chronologies which conform to a Jesus event around 30 CE, etc.
Obviously all of the above is cursory and I'm speaking in broad strokes, but all I can tell you is if you actually did research in this area in the original languages, it's really not a matter of debate. Even non-theist biblical scholars will say the existence of Jesus isn't a matter of contention (eg. Bart Ehrman).
Fwiw, the writings of Tacitus are far more compelling evidence than Josephus IMO. And that's not just me, but the scholarly consensus as well.