The traditional narrative about the origin of Old Testament writings states that the Old Testament texts were written
during the pre-Hellenistic period of the kingdoms of Judea and Israel, with later additions, and were mostly complete by the time of Jesus. Parallels between Jesus’s story and Old Testament texts occurred because the New Testament writers applied the Messianic prophecies in the Old Testament to Jesus.
I advocate the following point of view: up to the time of King Herod (40 – 4 BC), holy texts in Judaism were believed to have magical powers. They were only passed down orally, to save them from the risk of falling into the wrong hands and being abused. In this, the Jews were like the Celts, who are described by Caesar as having the same attitude: Bell. Gall. VI 14.
The Greeks were great writers but even they held back in writing down holy texts. Roland Baumgarten (Heiliges Wort und Heilige Schrift bei den Griechen, 1998, p. 223f) writes:
The Greeks appear to have made no comprehensive efforts to actually write down ancient traditional religious material. Cult practices were well rehearsed and preserved, apparently largely resistant to the imposition of written forms.
So the fact that the Jews did not write down their holy texts was not a specifically Jewish feature; it was the pattern in many ancient societies.