My Greek philosophy teacher would disagree with you. We know Socrates primarily through The Republic written by Plato, one of his students. So we have a direct eye witness account of Socrates' ideas.
Socrates has left us "the theory of ideas," the belief that true reality is the ideal reality and that physical objects are only shadows of the true reality. That is what his "Parable of the Cave" is all about. Basic to that theory is that only the best are fit to govern us which is why they occupy the most important positions in the society. And the best are to be governed by the Philosopher King who should be Socrates. This is determined by fate and nothing anyone can say or do makes a difference. For Socrates his ideas of justice were the most important things there were and he spent many years concerned with them.
Socrates did physically live in Athens but to him Athenian Democracy was the worst form of society because it was governed by plebians, lower class people who cared only about their appetites for more money, more status and all of the luxury they could afford.
And the example Socrates left supports this position. He neglected his farm and his own income despite the efforts of his wife to have him to attend to supporting his family. He preferred to live in his own ideal world conversing with his students. Today there are still followers of Socrates, intelligent people who have been seduced by those ideas. Some people have the means to live that fantasy life but many others of us do not.