You seem to have no idea of the timing of events relevant to the discussion:

1) When Taringail married Tigraine, he didn't marry into Mantear. Thus, he was still Laman's heir. Thus, his son, who was born at this time, was named a Damodred, to fulfil the idea of a brother and sister eventually occupying Andor and Cairhein's thrones.
2) Tigraine ran away, leaving Taringail her "widower". Taringail was still Laman's heir to Cairhein's throne.
3) Morghase won the Civil War, and married Taringail. However, she insisted he marry into Trakand. This happened before the Aiel war. Thus, when the Aiel war began, Taringail had been removed from the line of succession of Cairhein. When Laman died, this was the reason he didn't automatically become king. Jarna explains it in NS.
4) The Fourth War of Succession in Cairhein begins. We have evidence that Taringail did play the Game of Houses here (on Galad's behalf). However, with Taringail's removal from the line, and Galad's young age, and the Tower's support for Moiraine, there was sufficient dilution of support for Damodred that Riatin was able to gain the throne. This was not because Riatin had a better claim, but because they were able to enforce the claim better.
5) This ended Taringail's dream of a son as King of Cairhein. At this point, however, he was still heir to Hight Seat of House Damodred as the guide informs us. The real question is why he wasn't confirmed as such at this point. Could it be tepid support from Morghase? Could it be that other Damodred cousins did not want a "Trakand" to easily take the High Seat? Either way, the guide makes it clear that when Riatin got the throne, the High Seat of Damodred was still up for grabs.
6) Taringail was still plotting to become High Seat and remove Galldrian, when he was assassinated by Thom. This explains the wide belief that House Riatin assassinated Taringail to prevent further scheming from him.

The critical thing here is that when Taringail was married to Tigraine, he was not barred from the throne of Cairhein. That happened only when he married Morghase.

ETA: And here's the proof for that-



That's conclusive proof that Tigraine married into Damodred. Since Elayne would have been a Damodred if Morghase married into Damodred, the fact that Galad is a Damodred proves Tigraine did marry into that House.