All well and good except...
There are no age comparable documents that support this. Your post says it..."they think".
As you said, there was probably a man (not sure what he was called), who went around telling everyone to be nice to each other. That man had a few followers, then a few more, an underground movement started, remember Rome banned Christianity and the like, and the whole thing gathered momentum leading to his capture, death (no one knows how/ where either really) and martyrdom.
A few hundred years later, with most Roman soldiers now Christians, Constantine makes it official. You can worship who you like. He then chooses 50 stories and lobs them all together for his book. This is not the bible in any shape or form.
Rome and it's territories are Christian.
Then forms of the bible start to appear...
As far as churches thousands of miles away...the Roman Army is to blame for that. Christianity spread to all four corners of the then known world. The Army was mostly made up from people from the areas they conquered. They were then shipped elsewhere to fight for Rome. They took their religions with them.
I'm all for the archaeology finding proof one way or the other. It would be exciting either way. But I suspect we will never properly know unless a document is found. But, as I have said earlier, finding one that said he was a normal man, with a normal family and not the son of a God would cause the very foundations of the church to crash down.