I think you also have to ask why he didn't write things down?
I suspect it is because the intellectuals of his day (the sophists) were skilled at using paradoxes to turn all discussions inside out.
Because he couldn't write things down, he would have to start each of his conversations from scratch.
Still the Socratic Method of systematically questioning the foundations of reason has proven to be extremely valuable. It is sad that societies tend to persecute people who ask the difficult questions.