Despite him denying that he actually taught his followers anything and that they did everything on their own accord, what argument could be put forth that he was actually a teacher
Teacher derives from a word for index finger, in the sense of an aestel, reading a text to communicate it to others.
Educator comes from the root meaning 'bring out, lead forth'.
I argue Socrates was paradigmatic in defining philosophy, here: Weren't there any philosophers from Africa, America or the Middle East before Socrates? Philosophy is what Socrates did. And that Plato's Academy, and so academia, IS the fusion of the Socratic method and the Pythagorean math cult.
Is academia the same as school? I would say no. To pass a degree is write a dissertation, the equivalent of a guild's requirement to create a master-piece (albeit at either bachelor or master degree). To enter academia, means to gain a doctorate, to make a unique contribution with research. That is not being taught - it is being educated, led into the field, not filled with established knowledge.
And I would say the same of Socrates. An educator, not a teacher.