I'd be tempted to put down the scarcity of royal descendants in Gondor to the general mental state of the Stewards, as described by Faramir to Frodo.
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'Death was ever present, because the Númenoreans still, as they had in their old kingdom, and so lost it, hungered after endless life unchanging. Kings made tombs more splendid than houses of the living, and counted old names in the rolls of their descent dearer than the names of sons. Childless lords sat in aged halls musing on heraldry....
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TTT The Window on the West
The Númenoreans in general always seem to have had more of an affinity for the past than the future, going back to the son of Elros, Vardamir Nólimon, who was so enamored of ancient history he declined to accept his place as the king.
With a precedent like that, his descendants were set for their own preoccupations that clouded their judgement on the future of their race.
Regarding what you say of the Mouth of Sauron, I know of no evidence that he would have had elven blood. As you note, even the prospect of those of Dol Amroth having it is debatable, as it was a very singular trait.