That Socrates says "γνῶθι σεαθτόν, for an unexamined life is not worth living,” does not mean the life-affirming moral-purposive, ‘self-discovery’ claim of ‘finding oneself’ that everyone makes it out to be, whether or not the Existentialists (as my philosophy teacher proclaimed today) are inspired by the thought, because even if they were, they are not reacting to the modern pop-psychology-suffused, self-centered world views we are bred on in this day and age. I have a moderately inspired philosophy teacher who is quite mistaken, as far as I understand, to treat it in this way.
Regardless, it is most likely not even something Socrates actually said (γνῶθι σεαθτόν, that is)—and it seems the so-called quote only takes on any kind of gravitas because we are led to associate it with the man. Neither should my teacher make the vast and incorrect (in my limited understanding) leap that Socrates means, “there is no use asking about the motion of the cosmos because does knowing that make me a better person?” because the biggest error in arriving at this, is that that is not at all Socrates’ concern—the idea of people tempering and improving themselves to pursue ‘justness’ needs to be used in context, because first of all, there is no conception of the individual in ancient Greece as an individual; that is, he must train himself to be “just” and that is done solo, but he does not have an individual ‘will’ or ‘calling’ to be discovered, or a volition which can be thought of as a moral compass as it were, because, second, Socratic thought can only be called an system of ethical thought/a system of ethics insofar as it pertains to the maintenance of some kind of standard order or “justice” in the universe in this way; it may even help to think of it as a structural integrity, if you compare it to the similar Vedic system in ancient India.
Therefore, Socratic and other Greek philosophies are not relevant on an individual level in that sense. So the Existentialists, in talking about "the human condition,” appeal to a similar state of being emerging from/constituted by/spontaneously and randomly forming an 'order' or a ‘universal' or an organization of sorts, only insofar as it is something all humans are born into and struggle with, this “existence.” So this holds even when we concede that Existentialists are not concerned with this on the level of humanity but on an individual level, because the only resolution possible for them is an individual one.
Lastly, I struggle to accept the proposition that Socrates was a “moral philosopher” (my teacher says) in the way we seem to understand it because 'morality' as we understand it is, again, concerned with an internal damning and elevation—very Christian, in fact.