See, here is where I beg to differ. Asuka already understands that life hurts. She understands it and has accepted the fact that life can only hurt her. That's why she's suicidal. You are correct, if Shinji can only make her understand "Your mom's dead, you will always hurt, I can not make your pain go away, you are all on your own, only you can make yourself better", then she's worse off than before. Because that's precisely what she can't do. If she could be happy on her own, she wouldn't be in the situation she's in in the first place.
What Shinji CAN make her understand though is that life is not only pain. True, he can not shield her from all hurt that comes her way. He can not magically wave a wand and make her forget all the horrible things that she had to endure. But he can alleviate her pain. He can make her feel better, he can help her. Again, that's what they decide to do in the end: help each other. If the only way to not hurt is death, that would be futile and pointless. But death isn't the only way to not hurt. It's the only surefire way to shield oneself from further harm, thats quite a difference. Pumping yourself full of opium is one way to not hurt. I'm not saying it's healthy, but it's still true. In that moment, all the pain goes away. If it hurts more afterwards, that isn't a problem either if we truly accept that life always hurts.
It really just is a matter of perspective. From my point of view, if Shinji can make Asuka shift her perspective in a way that makes her not hurt anymore, or at least not as much, that means he did something to her and not that she did something to herself. Your point of view is based on the fact that Shinji can not force a certain outcome, but I do not agree that that's the case. After all, you can force people to be miserable, so the opposite has to be true as well. It is of course much more difficult and requires much more effort and time, but it isn't impossible, otherwise the concept of "help" would make no sense at all. So, rather than Asukas only option being to make herself better, the more precise statement would be that she has the option to accept having others make her feel better, if she can't manage on her own. Even if she doesn't want to accept that, her subconciousness can be influenced by others against her will to the point where it overpowers her rational thinking.
Take people with anxiety disorders for example. One way to help them is to force them to slowly face their fears, step by step. You can do that with their active participation or without, but it actually requires a certain forcing by the person helping. Another example is military training. You can mould people to do things that they wouldn't normally do. Again, willingness helps, but isn't required. Of course, success is never guaranteed, but saying that you can only help yourself and can only make yourself better divorces the inside world from outside influences and puts all the burden on the rational mind, ignoring the subconcious. Happiness and misery aren't a choice made by the rational mind. Change isn't a purely rational process. Even if you make the active and rational choice to change yourself in some way, for example becoming more orderly, what you are in effect doing is rewiring your subconciousness to the point where orderliness becomes a natural part of your psyche and not an active decision anymore, through the process of repitition. If somebody else forces you in the same way, the outcome is still the same. Of course, there is no guarantee of lasting success and depression is the best way to ruin progress, but depression is as much a decision as happiness is: Meaning, not a decision at all.
This all boils down to the concept of free will. You, and Evangelion for that matter, assume that we do in fact have it, but that's an unproven assumption. Your subconciousness makes decisions before your rational mind gets a chance to act - it only retroactively informs you of your decision. There is nothing in nature that would allow free will to emerge. Your brain is a physical object, it acts according to physical law. Even taking into account random quantum fluctuations in the brain, all you add is randomness, not the ability to actively shape the outcome. This means that your decisions aren't your decisions. They seem like they are, but they aren't. In fact, there are no decisions, only actions and reactions. Why it feels different is the crux of the debate. In essence, free will would require a non-physical soul or entity, able to act independently from the body it inhabits, unconstrained by any law of nature. Such an entity has not been found, and I suspect it never will be. Still, I like to delude myself with the notion that free will exists, in the same way that I just feel like I exist and the world around me is not only a figment of my imagination. A completely irrational point of view, based on unfalsifiable assumptions. Which just goes to show that we do not decide what we believe, which in turn makes our argument pointless: Shinji can not help Asuka, he either does or he doesn't, and Asuka can not help herself, she either gets better or she doesn't, something will happen but neither of them has any say in the matter.
It's still fun to assume that they, and we in turn, do, and I am happy to operate under that assumption. So thanks again for writing Genocide, it forced me to think, which is always a welcome occurence.