Taringail gets removed from the line of succession when he ,arrives Tograin, Galad gets no consideration as an Heir to either the Throne of Cairhien, or as HighnSeat to either House, despite his lineage. Making him a Damodred up front only make sense if Laman was gong to name him heir, which he clearly did not.

The question becomes why on earth would Andor agree to allow one of the Royal Line of Andor to be excluded in this way without a written treaty making him the legal Hier to the Sun Throne, just like Elayne put n place with Faile/Perrin. That obviously didn't happen since it boiled down to another House claiming the Sun Throne with no real push back. Galad would have been a fantastic choice for the AS camp, an aunt who is AS is the next best thing to having an AS ruler in place.

Damodred was in decline as Moiraine indicated when she discussed not wanting to be Queen of an unsavory House lineage. The fact that Taringail was not even considered as a candidate, nor was Galad during this time speaks volumes. If they had some kind of treaty in place that included Galad taking the throne of Cairhien then he would be the King already. By the time of Laman's death Galad must have been close to 10, which means when Tigrain vanished he should have already been named either Heir to Cairhien or House Mantear.