I think AC depicted him at a later stage in his life. He says in the game that he became fatter and less athletic with age. I heard he was a solider too, but most Greek citizens were soldiers (not the slaves, the citizens - BTW, it's funny how slavery is quite hidden in AC Odyssey, they've probably been scared of showing it, but in reality it was a pillar of their society).
I think Thucydides said that "The Nation that makes a great distinction between its scholars and its warriors will have its thinking done by cowards and its fighting done by fools." It's typical of classical Greece. Citizens were "complete" people who were supposed to excel in many domains.
Socrates is a real funny guy in the game, I like his trolling. In the 1st mission, he's annoying, but then his quests are very funny. Historians say that he would troll average people in the street like that, stating riddles or answering questions with questions. I like that.
As you say, he didn't want his teachings to be written (and there's a quest about it in the game). Plato is in the game too (as a kid).
AC Odyssey is real good. Real good. It would've been better, had the gameplay been more difficult (like in DS3), but it's still a great game.