Also a disclaimer: I think the existence of Jesus as a historical figure and his nature and status as a god or agent of god are two separate concerns, and I'm only arguing for the former. The same way I can accept the existence of Egyptian pharaohs without accepting them being literal gods.
The Bible can't directly prove the existence of Jesus, but the books of the Bible are evidence that people at the time thought Jesus existed, and that is good enough evidence to conclude that his existence is more probable than not.
No written document can be absolute prove of something, the Bible isn't special in that regard. None of the documents from that time can be "proven to be true", lots of them freely mix historical claims with supernatural ones, and even the ones that contain no supernatural elements at all could just as well be falsified for propaganda, profit, or even just for fun. Even archeological evidence can be dismissed in this way: following your reasoning, what does a stone bearing the inscription "PILATUS" really prove about the existence of Pontius Pilate? We can't know for certain who chiseled those letters into that stone or for what reason.
All of history is about the balance of probability. You can of course come up with alternative explanations, but are they more probable? Is there better evidence for them than there is for Jesus? The Christian movement had to have been started by someone, why should we conclude that it wasn't who people thought it was?
Paul talks about other Christian communities that weren't founded by him and don't follow him. Paul spends half his letters complaining about other Christians including Peter and James, the (supposed?) brother of Jesus, and the meetings and arguments he's had with them. He's defending himself from accusations, tries to argue for his authority and the authenticity of his revelation while trying to make up for not having been an original follower of Jesus, apologises for having persecuted Christians, etc. Is Paul making all of this up? If so, why is he not making up a more favorable story? If not, where did these stories come from? Are all of these people and communities made up? Is Paul made up? How many layers of conspiracy are there? And most importantly, is there evidence for any of that, and is it more likely than there having been a charismatic preacher who convinced a couple of people that he was a messenger of god?