I am well acquainted with Josephus, but also analysis that refute its credibility. I doubt you would not find such analysis at all convincing. But, if you would like me to present it, I can.
Tacitus isn’t describing Christ. He is describing Christians. He himself may have been convinced in the reality of a historical Jesus. But, forgive me (honestly), I consider that only as compelling as your own conviction. Which is certainly more than “not at all,” but it’s not nearly so definitive as, say, the presence of Napoleon in Napoleon’s tomb.
I would correct you in Tacitus's statement. He did indeed mention Jesus, though not his name "Jesus" specifically, he did say...
"The founder of this name, Christ [Christus in Latin], had been executed in the reign of Tiberius by the procurator Pontius Pilate..."
Which we all know from what the bible says, that Pontius Pilate had Jesus executed by demand of the Jews. I believe this directly states such. Tacitus would also have been around during the time the apostles were going around, which those apostles and other disciples had first hand witness account of Jesus's existence. Tacitus was accounting the time
You could pick and choose what you will. Your choice. People always do. For example, there are people out there that quite seriously believe the holocaust never happened. Crazy I know.
Josephus was quite spot on in the accounts of the Jewish-Roman war among other things. And he was around during most all the apostles ministry. While also being able to account for the miraculous things the apostles have done according to the bible accounts. I would highly doubt he lied about Jesus existence even if some of his other material is lacking. Especially in light with people like Tacitus, who quite clearly was talking about Jesus, mentioning the existence of Jesus ["Christ, had been executed in the reign of Tiberius by procurator Pontius Pilate"].
Those are just a couple of people attesting to the existence of Jesus. You still have Matthew, Mark, Luke, John (Jesus's cousin), James (Jesus brother), Peter, Paul, so on and so forth.
In Jewish law culture, you need at least two witnesses to testify with you. Jesus more than enough, even from secular writers, his family members, to prove he existed.
To look at it one other way, do you agree that these people existed?
Saul-Paul
John "the baptist"
Peter
James
Matthew
Mark
Luke
John [writer of book of John]
If you do, do you believe they also lied about Jesus's existence? Why would Christians and those who hated Christians all agree that Jesus existed? Three groups of people all account for Jesus's existence, Jews, Romans, and Christians. It would be hard to say they all are mistaken seeing as that even though they disagree with each other, that they all agree he existed.