If he was still Hier why did he not become King? Tigrain was presumed dead without an Hier well before the Aiel War so Taringail or Galad should have been first in line from your reasoning.

Taringail moved to Andor, Taringail become Prince Consort, Tigrain plotted to take the Throne from his second wife, all of which shows HE was the politically weaker individual. Makes no sense that the Queen/Hier of the most Powerful nation would make any political successions to a weaker nation and a monarchy clearly in decline (even prior to the Aiel War Damodred's days were numbered). Tigrain didn't love Taringail, Laman was insane and Modrelien was well loved (from what we know) ... Given that Taringail did not become King and Galad was essentially left as a ward of the state upon Taringail and Laman's deaths it is obvious that no written agreement to insure Taringail and Galad's succession was in place. Without that piece it's ridiculous to assume that Galad was meant to be a King by a treaty.

Facts that we know:

1. Taringail did not become King, despite Tigrain being long gone from Andor without a female Hier, why did he not revert to Laman's Hier when he was no longer married to a foreign sovereign?
2. Galad did not become King or High Seat
3. Andor is more powerful than Cairhien
4. Andor was even more powerful than Cairhien post Aiel War
5. Published intentions in WoT are ironclad ... Look at the value Elayne puts on this
6. Both of Morgase's children are Trakand
7. If Elayne were to die without issue Gawyn would probably be named High Seat of Trakand

Sum that up and we have no reason to believe that there was anything Published regarding the children born of the Tigrain/Taringail union ... which leads me to believe the union of Morgase and Taringail stripped Galad of several rights, including his right as Hier to the Throne of Cairhien and the High Seat of Mantear. Why else would he not have inherited one of these titles?