The evidence for the historicity of Jesus overall is far weaker than it would have to be to support anything like the Christian obnoxiousness that's infected the Western world for centuries. The Josephus passages are the strongest parts of that weak evidence, especially the "brother of James" one, but even that one is capable of being explained in a Jesus mythicist manner. Are you familiar with Richard Carrier's explanation of it?
When you talk about grasping at straws, you don't seem to realize that most questions of interest about the ancient world are settled by competing historical explanations that amount to straw-grasping, as it were. Without hard, archeological evidence, we're left with appeals to the best explanation, and there are often multiple ones that are more or less of equal value. It's the same in cosmology or in quantum mechanics, when we have no way of testing which interpretation is best.
Jesus's historicity has been mostly presupposed by Christians, but you can explain roughly the same body of evidence by assuming he didn't exist, too. Even if he did exist, he wouldn't have been the familiar figure from the gospels since those works are legendary at best.