That the Jews wrote their own testament is evidenced by historical references to later events, such as the victory of the tribe of Judah and particularly their first great crisis--the invasion of Babylon, which resulted in the destruction of Solomon's Temple and the Babylonian Exile. The stories likely circulated orally, but were not believed to have been written down until then, which is why they added or edited certain passages. (Though the Hebrew Bible/Old Testament wasn't finalized until the early Middle Ages, which is also how the Apocrypha came to be.)
The belief in a Messiah was a fervent one after the exile--God had promised David's heirs would rule forever, and the Latter Prophets (Isiaiah, Ezekiel, Joel, etc., who are working during the time of the Exile, so their writings are eyewitness, so to speak) gave many, many predictions of his return. For the Jews, the Messiah was a political, military and religious leader all in one, and he would deliver them from Babylon. Cyrus the Great, king of Persia who conquered Babylon and allowed the Jews to return to Israel, was even hailed as a Messiah, as it was simply a title that meant "anointed one," originally applicable to whoever God chose to carry out his will.
Christians believe these Messianic passages all predicted Christ. I can't dispute that--I read them, and that is what I see. Certainly, logic says that no one else has arrived to fit the description.
Some of the Psalms have been taken as foretelling Christ, but they are really just prayers and songs for all sorts of occasions. Psalm 22 is the most famous one, "My God, my God, why have you forsaken me?" It is certainly eerie, but I think it's our hindsight that makes it so, as if you read further it is about illness, not crucifixion. However, being a Psalm of lament and probably well known, it is fitting and haunting that Jesus should use it in prayer on the cross.
So, my rambling point is that these Messianic predictions have a long history in Jewish faith. I don't disagree they refer to Christ--I read them, and I see nothing wrong in interpreting them so. Holy men predicted them and wrote them down. God may have inspired the words, but He didn't write the Bible--Old or New. There wouldn't be editorial changes and repetition if He did. I don't see anything wrong in saying that.