I recognize the strength of the points you've made, and I'd like to clarify first that when I made that post I had yet to rewatch End of Evangelion. Having since rewatched it, allow me to take just a second to see if I can't do a better job of defending my points.
First, the hope. There is an overwhelming amount of despair in Evangelion, and that is, as we both agree, intentional. But the reason I think that End of Evangelion is so typical of the genre is because of the things said to Shinji while he is inside the Sea of LCL. Rei and Yui make a point of saying this is most certainly not the apocalypse. There is a lot of apocalyptic imagery at the end of the show, and with a red sea and a barren world, it certainly would lead anyone to conclude that Shinji is left alone in a world with somebody he hates, doomed to die and incapable of dying quickly.
However, both Rei and Yui tell Shinji that 1) He himself is capable of bringing everybody back and 2) Everybody else is capable of coming back of their own volition. In other words, this is only the apocalypse if Shinji wants it to be. They repeat the phrase "This is the world you wished for" numerous times prior to that, during instrumentality, and the finale is all about Shinji realizing that that isn't what he wants. In the TV finale Shinji realizes that he can change and that he can be happy, and in the End of Evangelion's finale Shinji accepts that he's okay with a world in which he will inevitably be exposed to pain and suffering, because he recognizes that there is happiness to be found within that pain. The implication of the ending isn't that Shinji is left to die in a wasteland, then, but that Shinji is at the New Beginning the title speaks of.
To that end, I feel that my assessment based upon the TV series' finale is fair, though I will readily consent that I am equally comfortable with the interpretation that this marks the beginning of Shinji's journey into adulthood, rather than the typical culmination thereof.
As to what happens immediately thereafter, I've given it a bit of thought since yesterday, and I confess I'm not completely capable of reconciling it, but if I were to do so, I'd say that in End of Evangelion Shinji's path toward the above revelations begins the moment Asuka tells him she won't help him and he decides to try to kill her. It's horrible and messed up, yes, but as Misato points out prior to that, it's still a valid symbol of Shinji taking a choice for himself and from that choice walking the path to the above revelations, because it was what Shinji wanted. When they arrive on the beach, and Shinji tries again, it's nothing more than a demonstration of what Shinji had already realized moments earlier: he's going to hurt and be hurt again, and he's going to question his decision to come back. But it's still valid because Shinji wanted it and made it happen for himself, prepared for once to live with the consequences. To that end, Shinji's motives and the aftermath of that spat don't matter. Just as the question of how Shinji will survive and how the world will go on turning don't matter, because what matters is that he's already armed himself with the emotional tools necessary to survive.
For this reason I feel it's typical. Shinji is confronted by a giant foe and enemy forces, collapses into himself, comes to an epiphany about himself, and from that epiphany draws the strength to end the crisis and move on with his life. It's possible I misjudge, of course, since half the beauty of the show is its ability to deceive the viewer into thinking that it is a typical Super Robot series so it can tug the rug out from under them, but I think that it's notable that the TV series ends with Shinji's being congratulated for his realization and End of Evangelion showing that realization granting Shinji the power to make the world his own. I think that that is very Super Robot. I think it's very much so in line with Mazinger Z's byline of "Becoming a God or a Devil" since, in point of fact Shinji is rendered capable of being either of those at the end of the series.
As to Asuka, I'm actually not completely sure how she gets to that beach at the end. We never see her entry plug destroyed, and there are a few differences in what she's wearing on the beach from what she had on when thrust back into the plug, but I suppose simple enough explanations for it can be appointed. I'm also not certain if she brought herself out of the LCL, or if Shinji brought her on his own. I readily admit that much about Asuka on that beach can be said to cast doubt on my conclusions.
I suppose, ultimately, that all of it can be boiled down to interpretation, though. As it is, if we're following the crazy theory that ReBuild is a sequel to Evangelion, who can rightly say what the ending means anymore, but when viewed as an independent and self-contained work, I think it's impossible to say that Evangelion ends on a sour note when Shinji is applauded for his realizations and speeches are given about the will of all life on Earth to live and be reborn.