"Messiah" means anointed one. The Jewish people are looking for the human person who is anointed to be as great a political ruler as David. Jesus insisted he was not this messiah. To back this up, Jesus said it was Moses (not Isaiah) who referenced him--Moses who lived long before David and Saul, in the time Judges--not kings--were first appointed.
For all we know, perhaps there will be another human Jewish leader appointed by God. This takes nothing from the task Jesus was anointed to do.
Jesus did indeed claim to be the Messiah. He may not have gone around yelling "I am the Messiah" but he did communicate it, more than once.
The woman said, “I know that Messiah” (called Christ) “is coming. When he comes, he will explain everything to us.”Then Jesus declared, “I, the one speaking to you—I am he.”Mark 14:61-62_____
But what about you?” he asked. “Who do you say I am?”Simon Peter answered, “You are the Messiah, the Son of the living God.”Jesus replied, “Blessed are you, Simon son of Jonah, for this was not revealed to you by flesh and blood, but by my Father in heaven.John 4:25-26_____
But Jesus remained silent and gave no answer.Again the high priest asked him, “Are you the Messiah, the Son of the Blessed One?”“I am,” said Jesus. “And you will see the Son of Man sitting at the right hand of the Mighty One and coming on the clouds of heaven.”Mark 14:61-62
There are more scriptures I could post, but it would take time to round them all up.
He also claimed to be one with God.
I believe one of the reasons the Jewish people rejected Jesus is because they didn't understand that the Messiah came as both the suffering servant, AND the conquering king. They were looking for the conquering king, a political ruler, so they missed the Messiah as the suffering servant, as prophesied in Psalms, Isaiah and other places.
Who do you believe Jesus is?
Christians shouldn't fiddle with Jewish writings and change their meaning after the fact. It does a disservice to Jesus. Isn't he enough without the dishonest embellishments?