Over the last few years I've thought about this a lot and decided that among other things, Evangelion is a sort of parable on the teachings of existentialism, or to be more exact, a defense of existentialism as an attack on the ideas of nihilism.
You're thinking too plainly if you think Eva is solely about human psychological flaws, the fragility of the soul, and odd Judeo-Christian symbolism. Sure it seems to be riddled with pessimism, and twice over depicts the destruction of man, but these are the things of which Eva is intentionally not advocating for. You have to understand that existentialism is not pessimistic and neither is Evangelion. Let me try to explain each of these ideas and maybe you'll see what I mean.
Nihilism is essentially the belief that all values are meaningless and that nothing can be known or communicated. It's often paired with extreme pessimism and radical skepticism condemning existence. The foremost advocate for Nihilism I can think of is Nietzsche who believed that one day it would cause a moral collapse of values, destruction of religion, and create a worldwide sense of purposelessness in everyone. (If you ever get a chance, read Nietzsche's Thus Spoke Zarathustra because, it is amazing). Nihilism is pretty extreme, and even from that small description I bet you can see that there's plenty of it eminent throughout Evangelion.
Both Nihilism and Existentialism hold that the universe and our present existence have no inherent meaning. Let me give you an example. Assume that time is infinite and the theory of the big bang is true. Now, because time is infinite, you could possibly believe that anything we do, think, or say has so little meaning within infinity that it means nothing at all. Let's take it one step further. The universe's expansion, contraction, and initiations of big bangs runs in a continual pattern; since time is endless, the universe as we know it is only one of the meaningless infinite occurrences of universes that ever existed and ever will exist. Imagine every combination of every arrangement of atoms in the universe each being put on a side of an enormous dice. Every time the big bang happens that dice is tossed and a differently arranged universe is created. Since the number of combinations is practically infinite AND the number of times each combination has been repeated is infinite, the world as we know it has no base upon which to make rational sense and furthermore has no inherent meaning.
This is where Nihilism and Existentialism split. Instead of taking the meaningless existence and destroying it, the Existentialist seeks to define his own individual meaning, despite existing in an irrational universe. Existentialism is about the individual defining everything (like the Democratic Party, lol). A true nihilist would believe in nothing, have no loyalties, and no purpose other than perhaps to destroy. Do you see the difference? One thing a lot of people don't understand is that existentialism is about living! This is because for a person to define themselves and their universe they must do so through the act of living. Because we live in a world without inherent purpose or meaning, we as individuals must learn to create it as meaningful based on rational choices of our own design. That to me is what existentialism is.
If you haven't seen it yet, nearly every emotional and psychological battle in Evangelion can be placed in one of these two categories of either Nihilism or Existentialism. To me it seems the whole series, in a way, is bent on converting an individual thinking in nihilist terms to thinking about the world from the perspective of an existentialist. This is done in two different ways, first by visually displaying the extreme horrors brought about by leading your life in a path of Nihilist destruction and self-loathing, and then, in the final two episodes, by giving you the alternate choice of perceiving the world from any perspective you wish to see it from. The series clearly resonates with this idea. Shinji is one who is unable to reconcile with the perception he has of himself and the existence which he lives in. On the edge of a downward spiral towards self-destruction, he is stopped by a simple recognition that he must create a sense of value for himself, and choose to exist in a world that he believes has meaning. This is extremely existentialist. You can literally take direct quotes out the series and find them somewhere in one of these two philosophies.
There are quite a number of other obvious clues supporting this. Take for instance the title of the 16th episode, [where Shinji becomes trapped in a Dirac Sea by Leliel (twelfth angel, the zebra striped one)] it's entitled "Sickness Unto Death." The name comes from a brilliant essay of Soren Kierkegaard's called The Sickness Unto Death. [Kierkegaard, being one of the foremost fathers of existentialism, discusses the nature and structure of despair (despair being the sickness unto death), it's causes and how everyone in some way shape or form is infected by it.] Other little things like this can be found all over the series, you just have to learn to look for them.
Whether the End of Evangelion fits into this pattern I'm not entirely sure, simply because you can easily look at the ending of the film from both viewpoints. It just depends on the way you interpret the final, like 10 minutes. For me I'm always going back and forth so I can't really tell you what I think (unless it's from both sides). That's how well put together The End of Evangelion really is, makes you keep coming back for more. It's terrifyingly wonderful.