The Romans have no record of this crucifixion, but then they wouldn't have; there were lots of crucifixions, and you couldn't keep track of all of them. There's no record in whatever Jewish documents remain either - no memo, say, from Caiphas to Herod: Get Jesus. The only record comes from Jesus' followers and they didn't start talking publicly about it until Pentecost, fifty days after the event. The first of the gospels, Mark, wasn't composed until about thirty years later. Astonishingly, each successive gospel became more detailed: Matthew, written about fifteen years after Mark, goes into a lot of detail surrounding Jesus' birth that Mark seemed unaware of. Luke recounts conversation on the cross. And John, composed last of all, tells us Jesus existed since before the creation, a point you would think odd for the others to overlook.
And yet, this minor execution to which neither the Romans nor the Judeans paid much attention, and which is recorded with remarkable inconsistencies in the only records we have, mattered a great deal, and people who heard of it were profoundly affected. In the end it didn't matter whether there were corroborating evidence or not, just as it doesn't even matter if Jesus were one man, or a composite figure made up of fifty would-be messiahs. The simple story, a man executed for preaching the word of love, who with his dying words, asked mercy on his executioners: "Father forgive them, for they know not what they do." Those words spoken from the cross compel all of us, Christian and non-Christian alike, to make a choice. You will either believe that Jesus was another victim of this crappy world of bully versus bully and the biggest bully wins until lesser bullies gang up against him, or else you believe that Jesus won, that his love was the victory, even to love the bullies who killed him. That you are presented with that choice, and that people of goodwill can choose to believe Jesus did not lose, but win, that the crucifixion was victory instead of defeat - is why the story matters, and why to this day the anniversary of the grisly and protracted death of a righteous man is still called Good Friday.