That is a reasonable interpretation, especially since Plato was not present in the dialogue between Phaedrus and Socrates, so the whole thing is either fiction

That is a reasonable interpretation, especially since Plato was not present in the dialogue between Phaedrus and Socrates, so the whole thing is either fiction or a dramatised retelling. However, this lends credence to the idea that Plato was being partly humorous about his own work, when he wrote down the section dealing with writing.