Could you share a source for this? The research paper I found has a different hypothesis; it links the slow transition to writing to trust, not an "old-school's attitude towards writing". Specifically the idea that the institutional trust relationships one formed with students, for example, would ensure the integrity of one's work. It then concludes that "the final transition to written communications was completed only after the creation of institutional forms of ensuring trust in written communications, in the form of archives and libraries".

So essentially, anyone could write something and call it Plato's work. Or take a written copy of Plato's work and claim they wrote it. Oral tradition ensured only your students knew your work; and you trusted them to not misattribute it. Once libraries and archives came to exist though, they could act as a trustworthy source of truth where one could confirm wether some work was actually Plato or not, and so scholars got more comfortable writing.

[1] https://www.researchgate.net/publication/331255474_The_Attit...