But upholding a local rulers (the sanhedrin's) sentence in a backward province? Why would they?
There are no Roman records from the Levant in the first century. No census records, no records of trials or crucifixions, no tax records, nothing. Its not just that Jesus is missing from the existing records; there are no existing records. Not only is there no official record of Jesus, there is no official record of the country he was from, the king who ruled that country, the Roman prefect who oversaw that king, the High Priest, the Sanhedrin, or anyone else for that matter.
There are Roman records from the first century, but most of them are from Rome and Egypt. A few unscrupulous authors like to claim that the Romans kept detailed records, and then provide a string of quotes from Roman records maintained in Egypt, before pointing out that there is no official Roman record that mentions Jesus of Nazareth. To call that disingenuous is an understatement
However, if we compile all of the works of literature produced in the first century, more of them mention Jesus than not. The New Testament alone contains 27 independent books from the first and early second centuries that mention Jesus, and he is mentioned by non-Christian (and non-contemporary first century) authors Josephus, Tacitus, and Suetonius. Josephus goes one further and describes the death of Jesus cousin-brother James, the same cousin-brother that Paul mentions within 20 years of the crucifixion.
Jesus background as a peasant from a humble family, the commonality of itinerant obscure-country preachers, the scarcity of contemporary writers and their focus on Rome, the problematic nature of Jerusalem and the Palestinian area itself garnering the attention, plus all the various other problems and issues in the empire of the time, would all serve to keep the focus off of Jesus while he was still alive. Even the teaching of Messiah-ship might look to outsiders like just more Jewish dogma
Now there was about this time Jesus, a wise man, if it be lawful to call him a man; for he was a doer of wonderful works, a teacher of such men as receive the truth with pleasure. He drew over to him both many of the Jews and many of the Gentiles. He was [the] Christ . And when Pilate, at the suggestion of the principal men amongst us, had condemned him to the cross, those that loved him at the first did not forsake him; for he appeared to them alive again the third day; as the divine prophets had foretold these and ten thousand other wonderful things concerning him. And the tribe of Christians, so named from him, are not extinct at this day.
The problem with the skeptics' claim is that no Testimonium Flavianum manuscript has ever been found that does not contain the passage. Testimonium Flavianum was written in the late first century.