The Jews didn’t want Jesus’ death on their hands, especially during Passover. So they tried to pass the responsibility over to Pilate. Although Pilate found there were no facts to condemn Jesus under, he sentenced him to crucifixion anyways.
The manner of Jesus’ death itself shows he was killed by the Romans, ostensibly for rebellion. (Thus the epithet, King of the Jews.) The Jews would have stoned him to death for blasphemy, but got the Romans to do their dirty work for them.
The Jews didn’t have the authority to kill anyone without the Roman’s permission. It wasn’t that they didn’t want blood on their hands during Passover, it was they couldn’t kill him without being punished by the Romans.
It’s like the adulterous woman they took to Jesus. It was a trap for Jesus. They knew that if they stoned her and killed her on Jesus’ word and authority, the Romans would arrest him. So they tried to get him to say that the law of Moses demanded that they kill her. His response “He is without sin may throw the first stone” was brilliant. Since none would claim to be sinless, and risk the charge of blasphemy and expulsion from their position, no one would throw a stone.