The Jews realy did not make a messiah out of Jesus.
Paul was a Jew. the earliest followers of the Jesus movement were Jews. The Gospel writers (for sure Matthew and John, and possibly Mark) were Jews. The idea that Jesus was the Messiah started with the Jews.
Pauls Judaism is debated,and the gospels were written by Gentile converts to the movement.
I showed how Paul's Judaism is not debated in another thread, so I won't touch it here. As for Gentile converts, that simply is not true. Scholars agree that Matthew and John were Jews. Raymond Brown, one of the foremost authorities on John, states that John was a Jew (or the community that created John).
Bart Ehrman, in his Introduction to the New Testament, also show that both Matthew and John were Jews, and I believe he ascribes Mark as possibly being a Jew.
Jesus was a failed messiah the day he died, and while alive he was not that popular or even well known, due to teaching in small poor villages.
He was known enough, and being known is not a matter of being the Messiah. More so, while most of Judaism would have seen Jesus as being a failed Messiah, other Jews in fact saw him as the Messiah, and redefined what the Messiah was.
According to the bible Jews killed Jesus, and that is the tradition the unknown authors wanted to be passed down. That is because he was a Hellenistic deity, Jesus only found fame after death in communities the exact opposite of the real Jews he taught.
According to the Bible, it was the Romans who killed Jesus. John may get close to suggesting that it was the "Jews" who did the actual killing, but the synoptics show that it was in fact the Romans.
More so, when John states that the "Jews" killed Jesus, he is not talking about all Jews. He is talking about a group of Jews. This is clear as John also accepts that Jesus and his followers were Jews.
If you look at the communities in which Paul went, there were all communities that had a considerable Jewish presence. And the earliest followers were in fact Jews. All scholars agree on this.