She's the one who believed Rand sat in Caemlyn as an usurper, had ousted the previous usurper she meant to deal with herself (to regain her reputation with the nobility and her throne) which made her furious (and in ignorance that she was dealing with a Forsaken and stood no chance to oust Gaebril with a hired army of CoL), and beside her perceptions of that situation drove all her decisions with Perrin later, placing herself secretely with him to spy and look for chances to help Elayne's cause.

Elayne was never angry over "the chair". She didn't like the fact that Rand had placed it aside to sit on another throne when she saw that in TAR (the perception that Elayne was furious with Rand over this comes from a POV of Demandred, where he misjudges Elayne's reaction, implying he thought could be useful as an enemy of Rand over his conquest of Caemlyn. he obviously didn't know back then Rand and Elayne had a personal relationship and Elayne was merely ruffled, not furious with Rand, to see the throne no longer in its rightful place). What sparked her "anger" with Rand (her displeasure/frustration at the situation would be more accurate. She understood right away what Rand meant, she was just frustrated at his misjudgement of the politics involved) is rather that he interferred in the succession process and announced he meant her to have the throne, as if he would put her on it. In terms of Andoran politics, this was a bad move. It made her look weak, and like a puppet of the DR. The situation was even worse than Elayne thought at the time, and the perpeption that she was Rand's creature merely one problems among many (manipulation of other claimants by the Shadow, Morgase's bad moves under Rahvin and the alienation of all her allied Houses and so on). The perceptions about Rand didn't play as much as it could have, in part because Dyelin had probed Rand in private and came to understand that what he meant about Elayne and the throne was different from how it looked (not making her a puppet, but because she was the rightful ruler... Dyelin was never made fully aware of the other aspect (though with her pregnancy she came to suspect), that Rand trusted Elayne as his ally, and why he did), and decided her to maintain her support for Elayne and work to help her by leaving the issue of her own claim up in the air until she'd managed to reconcile the ex-allies of Morgase to Elayne's cause (had she announced straight out and too soon she was renoucing her claim, Elayne was probably finished. She managed to manipulate Pelivar, Arathelle and co. - who wished her to be Queen - by not formally removing herself as a claimant until the right time).

I don't really see any possibility left for Rand to take the Lion or Crystal Throne. Since his epiphany, he's distancing himself from mundane politics, his plays are all for the world's established leaders to finally step up and take their responsabilities in the war against the Shadow, more and more seing himself as a "special agent" to deal with Shai'tan at the end, and an inspirational figure.

"Dark Rand" in his megalomania could have insanely tried to proclaim himself the rightful leader of the Aes Sedai, but the fact the men had been cast out and no longer Aes Sedai was well established in his mind by then and the notion never crossed
his mind to topple the Amyrlin Seat (mind you, it's rather post-epiphany Rand who stands a fair chance to plant openly the notion that the Asha'man should be allowed to become Aes Sedai and rejoin their sisters as equals, as a way to pay what he now perceives as a personal debt owed and wrong to the Asha'man, to have driven in them they are just weapons, expandable and doomed, New Rand now sees it was all about not losing sight and giving up on what you were figthing for. With what goes on at the BT, the name and look of the Asha'man will be associated with Dreadlords and Black Ajah soon.)