Shinji is often said to exhibit a kind of "real teenage weirdness." His sexual desires, his frustrations, and his futile rebelliousness are all a huge part of what separates Shinji from shitty clones like Shuu from Guilty Crown.
Just being a coward who occasionally rises to the occasion isn't enough to make a character interesting. Shinji is much more than that. This is what elevates Shinji, and the series as a whole, to the level of a masterpiece.
What matters is that we can identify with him, even if we are not exactly like him. He seems human, and this is important, especially if you are trying to tell a story that is as much about human psychology as anything else.
His angst and mood swings seem organic, rather than merely convenient. This is essential if we are to "believe" the character has a life of its/his/her own. I cannot over-state the importance of this element. Series that do not succeed in this regard are doomed to mediocrity.
When we start to see the strings behind the puppets, when the characters change artificially, that breaks immersion. We cease to believe in the character, and this makes us cease to believe in the WORLD of the character.
I believe that, were it not for the success of Shinji as a protagonist, the show would be remembered primarily for Gainax's technical achievements. None of the other elements work without a character like Shinji in the middle of everything.
In conclusion, I believe that the world of Evangelion traps and haunts us only because Shinji is a highly-effective conduit between us and that world.
