When you said we can know "quite a bit" about Jesus and that we can "reconstruct a picture" of him, I was hoping it would be more than a few basic facts eg. he was a teacher with a following who was crucified.

The whole premise of (much of) Christianity is that we can have a personal relationship with Jesus where we get to know him as a person. Even assuming the Bible is trustworthy, it doesn't tell you a whole lot about his personality or his backstory; it leaves one to fill in a lot of the gaps, probably with projection and fantasy, hence the huge varieties of perceptions of "who Jesus is" across different cultures and eras of history... and even within the same church.