The problem with that scenario is that both Tigraine and Morgase outranked Taringail at the time of their marriages. Taringail was Laman's nephew, not his son. Though Laman had no children at the time, it was presumed that he eventually would. And those children would be in the direct line of succession to the Sun Throne - well ahead of Taringail. But Tigraine was the Daughter-Heir when she married Taringail, while Morgase was the actual Queen. It would have been impossible for either Mantear or Trakand to hold the Lion Throne after taking the name of a foreign house - especially through a husband they outranked. If Morgase had taken the name Damodred after winning the throne, she would have immediately lost it again. Prior to the War of Succession, Taringail certainly outranked anyone from House Trakand (which is why a Cairhienin lord would have gotten the idea) and any such marriage would have gone that way. But "Morgase Damodred" would never have won the throne. And once she won, she was his superior in every way. It was considered good politics in Andor for Morgase to marry Tigraine's widower. It would have been political suicide to take his name.

I suspect he was a Mantear when he was born, but took (or was given) his father's name when his mother disappeared. Or at least when his mother's family lost its claim to the Lion Throne. When he was two years old. He could easily have been adopted by Morgase and taken the Trakand name, but for some reason that never happened. Perhaps his father's wanted him to keep the Damodred name to keep his chances for the Sun Throne alive (Taringail certainly wanted to have his children on both thrones). Or perhaps Galad chose it himself (once he was old enough) to honor his father. Or to keep a continual reminder that he was no threat to the Daughter-Heir's ambitions. Whatever the reason, his name meant that his younger half brother (one of them) would be the First Prince of the Sword. And he would be able to follow a different path.

What is this "Emond's Field" people keep mentioning? Mat, et al lived in a town called "Two Rivers," not a village called "Emond's Field." 😕 Seriously, Rafe couldn't even keep the name of their home village the same. The words "Emond's Field" appear nowhere in the show. Which makes one of the good things in the show - the song "Sing of (or Weep for) Manetheren" mean a lot less than it should.