I've been mulling this over since the last time I read this thread, and I think I've got something.
Even when he isn't so delusional that he thinks he is the DO himself, Moridin seems to think of himself as the supreme leader of the Shadow. When playing sha'rah, he doesn't think of the game as being between the DO and the Creator, the DO and the Pattern, or the DO and Rand. He thinks he is running both sides of the board, and is therefore guaranteed to win. Also, if you look at when this scene happens, it's not long after he's had that balefire run-in with Rand, and apparently it's right around the time that he is officially made Nae'blis, considering that that had to happen between Sammael's death and the last Forsaken get-together. So Rand has his crown of swords plus some other countries, and his "kingdom" is basically complete at this point. Sure, he has a lot to do, but the board is mostly set, Rand and Moridin each having just received the reins of an empire, or rather a set of pieces. To Moridin, this is great because he thinks he's the only player who is aware enough of the game to be able to move the pieces, and he is changing Rand's allegiances through guile.
But really, Rand doesn’t do a whole lot of side-changing, or any of that other Fisher piece stuff. I mean, you’ve got times where he has a little outburst of madness, but overall, he never really gives up hope. Every time he comes close, he has an epiphany or something happens to make him realize what he’s doing and stop it.
Compare that to Moridin, who is batsh*t crazy and does some incredibly weird stuff. He saved Rand from Sammael, and he's got some fairly decent reasons for that, but this probably benefited the Light-siders more than it did the DO. Also, with him being the DO's favorite, is it really that necessary to make sure Sammael snuffs it? If he primary motive is to die (which actually makes no sense given some of his actions), then why does he even care about being Nae'blis anyway? It might be a good reason for him to show up and watch, but to get involved? It could just be because he's crazy, or it could be Rand's ta'veren nature working on Moridin's shifting desires and unstable mind to move him like a chess piece.
And what about him telling Rand how to destroy the Forsaken? Why would he do that? Just to show off how smart he is?
Plus, where is he for most of the story, once he's resurrected? You kind of assume he's got a lot going on this whole time, but at the end it seems like he never really did. He apparently spends a lot of time playing games with himself and getting high on the True Power. For just a moment, I even thought that that was his big plan, that sucking down TP like that would allow the DO to attack Rand directly using his body or something. But apparently, this served no useful purpose to the Shadow whatsoever. Not only did he end up being more useful for the opposing side, but he did it unwillingly. He ends up getting played like a piece by Rand, who is supposed to be the blind cripple.
It worked out really, really well for the Light, though, didn't it? Maybe Rand could have used the TP himself, maybe, but there are several things that could have put a stop to that, such as if the DO didn't allow access or if taking in that much of the TP at once made him snap and go as crazy as Moridin.Also, the DO says that Moridin "brought you here." It's not really clear what that means. It seems like Rand could have touched that black stuff on his own, sans Moridin, but maybe he really was needed for that. That might also explain why Moridin was also frozen even though he wasn't part of the conversation- he was subsumed in either Rand's will or the DO's at that time. None of that happened when LTT attacked the DO.
By the end, however... Rand is the philosopher and strategist, whereas Moridin doesn't even know what the hell is going on. He decides to have a swordfight, because by this point he's totally snapped, partially from the TP, and partially because of his desire for what probably drove him to the DO in the first place- significance. Sure, he says it was just the smart thing to do, and Rand says that he was probably looking for power, but neither explanation completely fits Elan. If he joined for power, why wouldn't he just openly admit to having joined for the sake of power? Plus, whereas he seems to like having power, he doesn't use it very much. He is the leader of all the Chosen, but he rarely even bothers to make sure they are all on the same page. He does tend to show off a little bit around them, but he doesn't often assert his authority, as if he doesn't really care what everyone else is doing. Why was it so important to him to be Nae'blis if he never really cared to act like one? Probably because it's an important job, and whoever is doing it must be a really significant person, in a logical sense. He was already a celebrated philosopher, but the general public was more familiar with the scientific advances that came about as a result of his work than in the actual philosophy.
He also really enjoys knowledge and thinks he's entitled to know everything. Curiosity may have killed this cat. The DO may have actually offered him knowledge, or it might have been something almost accidental. "Some new, sinister power appears and no one knows what it is? It's threatening the whole world? Well, as the celebrated expert on the nature of reality, I had better go check this out. I'd better not tell everybody, though, because then I'd have to admit that I don't actually know what I'm doing. I'm going to be such a hero when I get back!" How easy would it be for the DO, doing stuff like what he did with Rand when he almost got him to agree to the destruction terms, to snap up somebody like Moridin, who craves significance and knowledge above all else, and also has depressive tendencies? So he goes over to the DO, and later even thinks he is the DO sometimes, probably because being himself, the moron who just had to go and spring the bear trap on himself, was unbearable to someone like him, who has to be the big force that turns the wheel, not just some schmuck who got himself caught up in the gears. That's what makes him such a perfect Fisher piece, that he wants to be victorious and important and doesn't seem to care very much about what side he's on or what the absolute final outcome is. He has to be on the winning side, and he has to be a driving force behind it.
As Rand becomes darker and, surprisingly, as he finds the light again, he is often compared (usually by himself) to Ishamael, as if he is experiencing a lot of what Ishy turned him against hope. This makes sense, obviously, as a foil for the hero, but it also kind of shows you how Rand and Moridin are switching places. Moridin becomes less and less sane, and a hell of a lot less feisty. Rand, who used to be a poor, confused kid without a snowball's chance in hell against his enemies, becomes more confident and composed. Moridin still appears confident, but he's definitely coming apart at the seams, and he does less strategizing when he should be doing more. It almost seems like he just gives up, though he does still show up for the Last Battle.
When Rand mentions how some of the Forsaken might never have changed over if they had been treated differently in the AOL, he obviously means Demandred (because he acted like a jerk to him and stole his dream woman), but he also definitely means more than one of them. Elan Morin was supposedly friends with LTT, but LTT was kind of an arrogant jerk, and Elan probably spent a lot of time cooped up in his own thoughts. The results of his works were celebrated, but hardly anybody actually read what he wrote. This was the kind of stuff he dedicated his life to, in a time when most people were into goofing off and having fun. Most people would probably consider him a Debbie downer and would politely shun him.Not much is said about Ishy's past outside of him being a philosopher, but it's hard to believe that he was ever a seriously happy guy. It seems likely he was starved for attention and admiration.
Rand, however, is raised in a great place with great people who teach him the skills he needs in life, like how to cope with despair. Ishy never learned that, which is probably why Rand seems a lot more sympathetic and even friendly toward Moridin near the end, even in that one scene in TGS. If it's not a Sandersonism, then his slight warmup towards Moridin as the story comes to a close probably comes from some recognition that he could have helped the guy before he became so unhinged and didn't, but it could also be that Rand has finally really understood what's going on, and he knows that Moridin is actually just a poor, crazy bastard who's overall a lot more useful to the Light than to the Shadow.
So really, in this scene, the board is laid out and ready to be played in real life, just like the one in front of Moridin, but he isn't a player. He is the Fisher, moving to the DO's wishes, and to Rand's, once Rand really learns how to play the game... which basically means he just had to learn the pieces work.
((Moridin did get one piece of all this right: either way, he wins. At least, if his real goal was to be a major figure in the epoch of eternity, he did that. He played a key role in defeating the DO, and was even cremated with full honors as a hero, getting that recognition that the world (LTT?) never quite gave him before. You might say that's his reward for serving his purpose in the Pattern.))