This is just a bit of propaganda to cover for the fact that it is the Roman Catholic Church, and to legitimise all the pogroms and persecution of the original and foundation religion - Judaism - of which the Christian cult is an illegitimate off-shoot. i.e. try to propagate hate for all Jews as the killers of the messiah. And they were very successful with the propaganda too!!
How the heck would you know what Pilate did or didn't want? The writers of The Bible were not some neutral source, they crafted a work of propaganda.
At the end of the day Jesus was (apparently) executed by Roman soldiers via a Roman mode of execution.
