Because of Galad's penis and matrilinearity, Mantear had no such thing as "ironclad succession" beyond Tigraine, and if Galad remained an only child, that the main line would have to let go of the High Seat to a junior line was nothing next to the distaster that House Mantear as a whole would lose the Lion Throne because it had run out of any female offspring who could succeed Tigraine. Mantear still had plenty of males relatives, the High Seat itself was never in danger.

Had Galad been her only child, he may not have inherited the High Seat anyway, it would have depended if a female offspring had been born in the House in the interim. Tigraine would have worked very hard to convince an allied Great House to marry a suitable daughter to Galad or to another close Mantear male relative such as Luc (a woman like Morgase's younger daughter, one of Luan's granddaughter etc. Anyone with enough lines to Ishara any House was willing to let go to Mantear in exchange for her becoming a Mantear Queen after Tigraine). That woman, more likely than Galad, would have inherited the High Seat of Mantear. Of course, that would not have been so easy, as the Houses stood a chance to gain the Throne for themselves if they refused to marry a daughter to Galad.... but if they refused, another House might accept Tigraine's offer....

Galad's age when Tigraine died is what ruled him out of her succession to the High Seat, not his name.

Remember, we don't even know for sure that just because Galad adopts the name of Damodred he was excluded under Andoran law from his mother's House. He may well have been part of Tigraine's house despite the name, in fact he most likely was as it would be very strange a noblewoman could legally strip her eldest child from his birthrights. Taringail joined Morgase's House officially, and yet retained his Carhienin name, so there's no legal conflict in doing that. Only Tigraine's death and Morgase's adoption possibly has changed Galad's Andoran house (if he's part of one, but he most likely is), but he still uses his Cairhienin House name. He might just follow his father's wishes in that, for all we know, not matter what his Andoran house is.

That Galad remained a Damodred by name counted for not much. Mantear lost not much, except Laman may have made it a sine qua non condition that Taringail's first son would be allowed to retain the name of Damodred, to emsure Taringail a descendance, perhaps in accordance with some Cairhienin law, for all we know (Elayne's case seems to indicate it's not even necessary - she could use the name Damodred in Cairhien and Trakand in Andor if she so wished - and it might rather have had been a matter of pride for Taringail and Laman, pride in the name, not to see the larger part of Taringail's Damodred inheritance pass to a Mantear or Trakand and so on. He passed some to Gawyn-Elayne, but no doubt his eldest got the lion's part). The fact he was Tigraine's son was enough if he had to play a role in Andor eventually. He would have been his sister's First Prince and all that.

Male genealogy counts for fairly little in Andor. It's not even unknown for a Queen to marry a commoner. Ishara herself did just that. That it isn't done more owes to the fact wedding powerful lords into your house or your minor daughters into other Houses make solid alliances which any House needs to maintain their power and eventually advance in position.