And since Morgase had no brother it's possible there is a contingency for FP.

Also, Elayne tells Gawyn his duty as FP includes being Captain General, but it's clear that there are other duties since she published that he was free of them at her will.

The point I am making is that Taringail should have succeeded Laman and Galad after him once Tigraine vanished. Since these things did not happen is seems like they had to renounce Damodred inheritance. What follows is that when Taringail married Morgase Galad may have been reverted to Damodred so that a Trakand Queen didn't control a Mantear High Seat. And since Laman wasn't even in consideration for the Sun Throne after Laman's death it sounds like there must have been some provision in place to exclude him.

Do the math. Galad is born of the Royal Houses of Andor and Cairhein, yet inherits from neither side upon the death of either parent and doesn't even have the name of his matrilineal House which in Andor would carry more weight for him since he did not inherit anything from Damodred. That would make some sense if he had inherited the Sun Throne or at the very least High Seat of Damodred, the mystery is why Galad is marginalized in both Andor and Cairhien. He has birth right to both a Throne and a powerful High Seat, yet got neither and is essentially left to the mercy of his adoptive mother. To your earlier pnt Elayne may be able to hand him the Sun Throne, but his legal right seems limited for an unknown reason.