I think Christy has already given a good answer which I chime in with as well. I’ll even add a bit (for me) that obviously Jesus mythicism would scuttle my faith, but even something a bit less silly than that: If early scrolls were discovered tomorrow that made it clear that the early disciples were not in fact faithfully repeating what Jesus had taught them, but were instead “inventing” the bulk of his sermons and teachings - making up their fledgling religion to rescue it from the tatters of the crucifixion event; this too would be a death blow to my faith. Of course such a discovery would coincide with the demise of resurrection evidence too, so that answer is still tied in with this.
Note: By this I don’t mean that the presence of discrepancies, exaggerations, different-point-of-view issues that will always be present with multiple or long human accounts worry me much. So I most emphatically reject the approach of those whose “faith” is so brittle and fragile that they need a perfectly dictated “holy book” that will stand up to their modern demands and notions of what such a book should look like - obliging themselves toward whatever gymnastics are necessary to explain away the textured humanity found in the accounts. While I may have emerged from such an outlook, I emphatically reject it for the bibliolotry I recognize it to be now. What I’m saying is that if the thrust of the central message(s) of Jesus teachings were all just made up on the fly, I don’t think my faith would survive.
The accounts of the appallingly embarrassing failures of the disciples militate (for me) against the possibility that they were the originators of the entire story. Did their own biases creep in there as they (and/or their own later followers) got their words down on paper? Sure. Maybe even an extra story or two (such as the woman caught in adultery in John’s gospel). As long as those things aren’t contrary to the spirit of who Jesus was and what he taught, those things don’t bother me over much, and I’m willing to see them as incorporated into the narrative of “who Christ is and was” and “who we are in Christ” all under the authority of that very Spirit.