I guess Christians would care whether there was a historical Jesus, but in many other religions, the philosophy speaks for itself. I don't care if there was a historical Socrates, for instance, we have the words, and it's the ideas that count. I do think Christianity works for Christians whether or not Jesus was real. Their "having a relationship with Jesus Christ" is all in their minds.
It works for philosophy, yes, but in Christianity, the physical existence of the Messiah is required. His birth and death are both requisites for this new pact we're supposed to have with God. Without him, it means nothing. In other words, if we were to prove beyond any doubt that tomorrow that Jesus did not exist, then Christianity would collapse.
As to whether Christianity could begin without Jesus, it did begin without Jesus. Jesus was a Jew, not a Christian. If there wasn't a Jesus, it could be that Christian legends are based on another person, or an even older myth, or based on the stories of several people that were combined into one over time. I don't think there was a Moses, either.
I didn't mean it that way. I meant "was Christianity based on a myth, or was it based on a real person (around whom perhaps a myth was created)?" But you've answered the question well enough.
I tend to think there was. He's not well accounted for in history, but most people from that far back are not. There are clues, however, and one such is Luke's invention of the census, of which he gets every possible detail wrong. He does this to put Jesus in the town of Bethlehem, because doing so links him to David and thus fulfills the prophecy. Matthew fabricates an entirely different story, but the result is the same, with Mary ending up in Bethlehem. If Jesus was indeed a mythical figure and not a real person, why not simply make him from Bethlehem? The need for the lie hints that someone real was at the center of these stories.