If your only requirement is that a man once existed by the name of Jesus and was crucified, then the bar is on the floor. Jesus was not a rare name, and the Romans crucified many, many people. It is not out of the realm of possibility that these two common data points would overlap and give us a crucified Jesus.
Is there proof that it was THE Jesus though? Do we have corroborating evidence of a man travelling the countryside with his posse, changing the minds and hearts of the masses?
Do they need to?