It would not be a matter of Jesus “needing” to be literate, (to the point of deliberately making up false claims), so much as simply portraying him in a way that presumed he could handle any situation–which would include being educated. Someone who can read (and write?) simply has an advantage in the world that an illiterate person lacks, (particularly from the perspective of the authors of the gospels who were, themselves, literate).
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OTOH, Dio, believing firmly in his own favored scholars makes an absolutist case for illiteracy that requires a number of assumptions, as well. The case for illiteracy based on a lack of time or materials has a certain persuasion, but it is not the clear cut case that Dio would like to believe. There is enough commentary from people in the first century B.C.E. through the second century C.E. about the general level of education among the Jewish people that a more cautious scholar might wish to temper his or her certitude.
Among the tantalizing points:
When the Pharisees briefly held power in the early First Century B.C.E., they passed a law demanding that every father ensure that his sons could read and write.
Later, a further law was passed that actually talked about establishing schools.
These are not conclusive. Many laws are held in the breach more than the enforcement. It is also possible that the laws were only felt to apply to the merchant class and large land-holders with the poorest peasants simply ignored. Further, there is no reliable evidence that the required schools were ever built.
Both Philo and Josephus commented in their respective works regarding the Jewish people regarding the widespread, (pretty much universal), education among the Jews.
These comments are not conclusive. The language regarding the material that was taught was not explicit in naming reading and writing among the subjects. An understanding of the Torah could have been accomplished in an oral environment of memorization and verbal explication. And, again, they may have been discussing only merchants and land-holders while ignoring the larger numbers of peasants.
References to Jewish learning also appear in a number of Roman authors. (I don’t recall any similar comments from Greeks, but they had been out of power by more than a century by that time.)
Again, the comments do not explicitly address literacy and there is always the question of whether the peasants were included.
In addition, there is a presumption that the “illiteracy” model makes that may be quite valid–or might be unsupported in fact. Much is made of the costs of the materials required to produce a Torah and further materials needed to study. However, there does not need to have been a Torah in every home. One in each village (or even cluster of villages) would have been sufficient to provide a basic understanding of reading to children old enough to walk to the village but too young to effectively contribute to the labor force. Chalk and slates or sticks and hard packed sand are not that expensive. Children in Ireland were educated in “hedge schools” for several generations under similar conditions of poverty with the additional threat of British authorities seeking to break them up.
None of this is affirmative argument in favor of widespread literacy; it simply points out that the obstacles of time and materiel were not insurmountable obstacles.