There is the evidence that the Jesus story began around the time when the real Jesus is meant to have lived. So, why then? Who or what was the catalyst?
It's slightly more complicated, more interesting, worse for one version of a historical-Jesus case, and better for another version of a historical-Jesus case, than that.
The only details that ever made anybody think Jesus was born around 1 and preaching around 30-33 are the Census in one of his infancy narratives and the name "Pilat". And we already know the bit about the Census is wrong because that didn't work the way the Bible says it did, wasn't ordered by who the Bible says ordered it, and serves an obvious purpose of enabling the author to say he was from two different places in order to claim he satisfied two different origin requirements. And based on other sources' descriptions of Pilat, he wouldn't have acted the way the guy in the Bible does.
So, if we take the Census and the name "Pilat" off the table, when would we think the story was supposed to have happened, based on the remaining clues?
Historical-Jesus debates tend to lump the whole mid-00s together as one era, an era in which many Jews were thinking about all-out rebellion against Rome, a few leaders were trying to get one started while mixing the concept of that rebellion with Jewish theology, and Rome was squashing those movements and crucifying their leaders. The word for some other guys getting executed along with Jesus is often translated into English as "robbers" but elsewhere refers to rebels, not just thieves. Jesus himself is said to be thrown in with "the other {robbers/rebels}", and we all know that the "rebel" profile fits him while "thief" does not. But there are no secular sources saying that's what the setting was like in the early 30s. They say it was like that in the 50s and 60s. Before that, there's a period in which there no signs of Jewish rebelliousness or of the Romans going around crucifying people, because the people they would end up crucifying, the rebels, weren't "a thing" yet.
But it's more specific than just the general mood of the place & time. There are a handful of events that are mentioned in both the Bible and secular sources that are weirdly coincidental if they're supposed to have happened decades apart but make perfect sense as just one event being reported in both places. For example, Paul mentions a bout of social unrest which was kicked off by a guy named Stephanus getting attacked by a mob (I think that might even be what he says somehow inspired him to go around "oppressing" certain people). Josephus also mentions similar unrest in response to a Stephanus getting attacked by a mob. But Josephus puts it in the 50s or 60s (I'm not sure when exactly), and conventional Christian dating would need it to be in the 30s. And there are a few others like that. And during that same period, there was also another Herod as "king", and a different guy in Pilat's job whose style better fits the style of the guy in the Bible. And that era is not just the time when wandering preachers in general were around, but includes a specific one Josephus never names, called "the Egyptian", whose story sounds like a secular description of Jesus without the name
In short, pretty much everything else about the story says it needs to be set two or three decades later than most people currently think. That's when everything else lines up between the Bible and secular sources, except for the Census that we know the Bible gets wrong and the name "Pilat". If the Census hadn't been thrown in and the Roman guy's name hadn't been changed to Pilat, nobody would ever have doubted that the story happened in the 50s/60s and Jesus was "the Egyptian".
