That we have no record of Josephus writing about Jesus is not controversial at all - we do not. "The idea" that he did may be appealing, but it is not supported by available facts. The earliest available writings of Josephus about the time period have no mentions of Jesus. Later copies of the same writing get more and more elaborate descriptions of Jesus. All this provides is "the idea" that Josephus might have written something which was then cut away in the earliest copies of his writing we have. You enjoy invoking Occam's razor. Please apply it here.
Drawing on the claims on who Jesus was, he was an allegedly much more important man than a mere ruler of men. There would not be statues and coins, but there would be writings - lots of writings, from people who actually met and saw him, and were around him. And from Romans who noted what happened, and how he was executed, and the manner in which it happened. Pilates would have written to his superiors to clear his name, and this would be recorded. There would be immense amounts of documentation of such a momentous person.
As to the gospels, I dismiss their status as primary sources. They do not fill the criteria for those. None of them do.
And how do you know which gospels are "the false ones"? Can you please provide the selection criteria you apply? This is a serious question; no-one has ever managed to explain how they know which ones to trust and why, other than "I grew up trusting these, so I trust them", or "tradition says ...".
And yes, people write about and die for imaginary persons all the time. Christianity is not the first nor the latest instance of this, and the fact that people do is evidence of nothing at all.