What I am saying is that I find it a poorly worked out scenario filled with bad decisions. Galad as a Damodred did nothing for the Mantear line without Laman guaranteeing him the Sun Throne. It's the only rational reason the Mantear Royals should have allowed for Galad to be a Damodred rather than a Mantear. In effect Modrellien allowed her line of the Mantear family to die out by not having Galad inherit the High seat when Tigrain vanished. This potentially weakened Mantear for future successions since they track back connections to Ishara which Galad's line of the family in theory would have more of than other Mantear lines.

And no way Taringail marries into Trakand if he is Hier to the Sun Throne. I suspect something else went down between Tigrain vanishing and Morgase becoming Queen that at least put doubt on the succession to the Sun Throne and clearly had Taringail in an uncertain enough situation to renounce his lineage and potential for the Crown in order to marry Morgase as a Trakand, probably hoping Laman would eventually name Galad Hier if Laman saw political advantage in the Trakand union ... Thus leading to his hope of his children each being Monarchs. The war and assention of Riatin naturally changed that and probably set up his own demise for playing the Game in an attempt to claim a Throne for himself.

I believe we are missing quite a bit of information on how Galad ended up with a "standard" inheritance. I suspect the prequels planned for between New Spring and EotW would have addressed this more closely as it plays a large, albeit tangental, role in the development of the series and we know Moiraine had been in Elayne's presence at some point before EotW, but likely well after Taringail's death since she says she could sense Elayne's potential which likely means it was within the same year as her journey to the Two Rivers. Not to mention Moiraine doesn't exactly love her family so I can't see her dropping in to see her half brother at the Caemlyn court.

Regardless I find it foolish that the Mantears shuffled a member of their Royal line to a foreign House rather than make him High Seat when his grand mother died. Consider if Galad had a daughter and Morgase died without female issue ... Galad's daughter would have been a very likely candidate to succeed, but if she was a Damodred she would be ineligeable.