He was Laman's heir till he was forced to marry into Trakand, which automatically excluded him from the Cairheinin succession.

When he was married to Tigraine, his dream was to have a son on the Sun Throne and a daughter on the Lion Throne. Dyelin told us this, and this would possibly have been achieved by a treaty similar to the one Faile and Perrin signed (where it was stated that if Faile assumes the Throne of Saldea, one of her children will be named heir of Saldea, and another named Steward of the TR, so that too much power doesn't enter into one hand, and thus destabilize Andor).

Galad, therefore, was still a viable her to Laman when Tigraine ran away and Taringail married Morghase. However, to alleviate the loss of power that arose due to Taringails removal from the line of succession, Laman began a series of moves that included cutting Avendoraldera for a throne. That led to the Aiel War, and the loss of face for Damodred, and Laman's death.

Once Laman and his brothers were dead, the Tower immediately plotted to put Moiraine in the throne. Taringail probably wanted Galad to take it, and likely plotted for it, but wasn't successful because of the amound of power Damodred had fritted away recently. The Cairheinin also probably didn't want a boy-king who would be under Morghase's influence, nor did they want Taringail as reagent, since his marriage into Trakand was the root cause of the nations downfall anyway.

Morghase also didn't want Galad gone, presumably. She genuinely loved him, and sending him to Cairhein as heir to House Damodred would have meant he would be a pawn against the new ruler. The Tower probably had Elaida influence Morghase here to keep Galad out of Moiraine's way too. Once it became clear Riatin would get the throne, Galad would only have served as a pawn to other Cairheinin houses, something Morghase wouldn't accept for an eight year old boy. Galldrian probably gladly gave in to her demand to adopt him, to secure his throne further. The WT also likely encouraged this to gain favor with Galldrian, and to preserve stability in Cairhein.

Thus, despite being the best candidate for the throne and the High Seat of Damodred, Galad ended up second in line to be First Prince of the Sword of Andor instead. And thanked his step-mom for it.

More later.