Year 0 in Judea (and Rome, Damascus...) was a pretty literate period and literacy was (the story of jesus confirms) already a religious requirement... bar mitzva or an ancestor of that custom. Bar mitzva translates roughly to "eligible to uphold commandments." You need to read for that, or so the custom implies.

I'm sure that most of the religious/rabbinical tradition was still oral, and that rabbis did a lot of oral teaching. But, I think high fidelity oral "technology" was already heavily diluted, especially in judaic culture. Scriptural worship starts very early. It's hard to say when exactly, but it had to have happened while hebrew/canaanite was still spoken in the region. Aramaic had overtaken hebrew circa 400-500 BC. From that point, high fidelity transmission was done with writing.

Also, the multicultural/multilingual/multiregional context makes it unlikely that a high fidelity oral tradition existed in early christianity. IMO, the new testament was almost certainly compiled from earlier written sources... and oral telephone. I mean, the new testament isn't even in the same language as the sermons.

The same can be said about mishnah/talmud... the jewish contemporary to the new testament. Traditionally, it is seen as a compilation of jewish oral traditions, received at mount sinai and maintained with fidelity for two thousand years. Realistically, there were earlier written versions of (eg) Rabbi Hillel's teachings available to the scribes who compiled the Mishnah.

The traditional reasoning for writing the "oral torah" (resulting in talmud/mishna) was that oral traditions were dying, and that writing was necessary for fidelity. Multiregionalism, multilingualism and such were to blame... and the christians would have had even more of those problems. Fewer, more dispersed. No institutions. No common language. No old traditions. It's possible that there was a tradition of reciting Jesus' sermons, but that would be kind of culturally out of place. I think it's pretty unlikely. If there was, I think the new testament would have been compiled in aramaic.

Also... there are quite a few convergences between new testament stories and other (broadly termed) rabbinical accounts from the period. John the Baptist has his own religion, for example, and in their books you get some of the same stories, but with John in the Jesus role. I suspect there were many others, but have no modern adherents.

Judaism of that period aggressively trimmed out any new or recent "revelatory" writings. New prophets, new conversations directly with god. That's what many of the "apocrypha" are. From then on, religious scripture needed to be wisdom received from oral traditions, old sages and stuff. No revelation.

These, to me, strongly suggests late 2nd temple judaism was no better at oral tradition than us. That said, you can have conceptual fidelity without having word-by-word fidelity. When Jesus paraphrased Hillel, he was reaching across 400 years of oral (probably/mostly) tradition. Christians doing the same thing 200 years later probably had that level of fidelity.