By the time of Jesus' crucifixion the Jewish Sanhedrin had lost the power to perform capital punishment without Rome's approval.
This did not mean they could not execute criminals; it meant they had to work within the confines of the Roman legal system. This is demonstrated in Paul's trial before Felix in Acts 24, and addressed specifically in John 18:31
Then said Pilate unto them, Take ye him, and judge him according to your law. The Jews therefore said unto him, It is not lawful for us to put any man to death:
Stephen was executed illegally by a mob, James the Lord's brother was illegally executed during a Roman interregnum. These circumstances gave hostile religious leaders the opportunity to break the law without being punished by Rome for doing so, but their actions were not legal.
They may well have considered trying again to stone him
Some consider Sanhedrin 43a to preserve an arrest warrant for Jesus, in which it is indicated that Jesus was to be arrested for sorcery and put to death by stoning. Historian Paul Maier examines the evidence here and suggests this was originally written before Jesus' death (since afterwards it was well-known that He died by crucifixion).
In any event, whether there had been an intention to execute Jesus themselves (which would have meant death by stoning) or not, this is not how it played out.
Why death by crucifixion instead?
Talmage suggests both a theological & political significance to Jesus being put to death by crucifixion instead of by stoning:
John the apostle intimates...a determination on the part of the Jews
to have Jesus put to death not only by Roman sanction but by Roman
executioners; for, as we readily may see, had Pilate approved the
death sentence and handed the Prisoner over to the Jews for its
infliction, Jesus would have been stoned...
whereas the Lord had plainly foretold
that His death would be be crucifixion, which was a Roman method of
execution...Furthermore, if Jesus
had been put to death by the Jewish rulers, even with governmental
sanction, an insurrection among the people might have resulted, for
there were many who believed on Him. The crafty hierarchs were
determined to bring about His death under Roman condemnation. (Jesus
the Christ pp. 632-633)
I can see at least three reasons why the Sanhedrin preferred to have Jesus put to death by Rome:
Fear of the people. The multitudes could be manipulated but could also be unreliable. If the Sanhedrin sought to turn the people against a popular figure, they risked the multitude turning on them instead! They may well have generated an uprising, but risked an uprising in the wrong direction. By having the sentence decreed and carried out by Pilate, they ensured its success. If the people rose up against Rome, Rome had the resources to brutally quash the rebellious.
That this factored into the Sanhedrin's plan is evident by the fact that they arrested Jesus in seclusion at night, rather than taking one of many opportunities to have Him arrested when He could easily be located while teaching in public during the day (see Luke 19:47).
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- They knew Pilate could be manipulated into doing their dirty work. Despite the fact that Pilate wanted to let Jesus go, his own standing with Rome and the risk of an appeal to Rome were adequate to paint him into a corner with this threat:
And from thenceforth Pilate sought to release him: but the Jews cried
out, saying, If thou let this man go, thou art not Caesar’s friend:
whosoever maketh himself a king speaketh against Caesar. (John 19:12)
The significance of this threat is explored on this site here, and my own research on the political realities Pilate faced if he did not comply is found here.
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- A slow and painful death. The Sanhedrin did not merely want to kill Jesus, they wanted to destroy His movement. Having Him put to death by crucifixion would be humiliating, and having Him put to death by Rome would intimate that His teachings were opposed by the might of Rome. The slow, tortuous death by crucifixion also afforded ample opportunity to mock Him as He suffered (see Matthew 27:39-43).
Conclusion
Death by crucifixion would not only be more humiliating and more painful, but by having Jesus executed by Rome, the Sanhedrin could accomplish its designs with the fewest risks (in their minds) that He would escape yet again.
The relevance of Passover to their deliberations is summarized in Matthew 26:
4 And consulted that they might take Jesus by subtilty, and kill him.
5 But they said, Not on the feast day, lest there be an uproar among the people
The Passover crowds gave the Sanhedrin incentive not to cause an uproar, but they clearly had separate motives for wanting Rome to do the dirty work, regardless of the date.
Addendum - the references to the Sanhedrin in this post can be correctly read as the majority of the Sanhedrin. While it is clear that the convened Sanhedrin collectively favored the execution of Jesus, this does not mean all members of the Sanhedrin shared this view.
I reference the term Sanhedrin specifically to avoid the vast oversimplification that the Jews in general wanted to kill Jesus. While some individual Jewish people did want this outcome, it was religious leaders, not laity, who organized the execution plot, and it is worth noting that the people who hailed Jesus at His triumphal entry were also Jews. There was not just one single Jewish viewpoint on Jesus--who was Himself a Jew.