Plato would not have missed the irony. He was incredibly literary and ironic in his writing.
There’s always tension in Plato as to what was his opinion and what was Socrates’. Furthermore many characters in his dialogues are explicitly meant to be wrong in their opinions. Just reading a Plato quote as evidence of Plato’s actual opinion is a route to confusion.
This idea as communicated in Phaedrus, by Socrates, was certainly Socrates’ opinion: Socrates never wrote anything down.
