I think Tolkien himself gives the answer to the problem of places of worship in his letters.

#143: "That is why I have not put in, or have cut out, practically all references to anything like 'religion', to cults or practices, in the imaginary world. For the religious element is absorbed into the story and the symbolism."

#153 (footnote on p. 193): "There are thus no temples or 'churches' or fanes in this 'world' among 'good' peoples. They had little or no'religion'in the sense of worship. For help they may call on a Vala(as Elbereth), as a Catholic might on a Saint, though no doubt knowing in theory as well as he that the power of the Vala was limited and derivative. But this is a 'primitive age': and these folk may be said to view the Valar as children view their parents or immediate adult superiors, and though they know they are subjects of the King he does not live in their country nor have there any dwelling."

#156: "... and established a kind of diminished memory of Númenor in Exile on the coasts of Middle-earth – inheriting the hatred of Sauron, the friendship of the Elves, the knowledge of the True God, and (less happily) the yearning for longevity, and the habit of embalming and the building of splendid tombs – their only 'hallows': or almost so. But the 'hallow' of God and the Mountain had perished, and there was no real substitute. Also when the 'Kings' came to an end there was no equivalent to a 'priesthood': the two being identical in Númenórean ideas. So while God (Eru) was a datum of good Númenórean philosophy, and a prime fact in their conception of history, he had at the time of the War of the Ring no worship and no hallowed place. And that kind of negative truth was characteristic of the West, and all the area under Numenorean influence: the refusal to worship any 'creature', and above all no 'dark lord' or satanic demon, Sauron, or any other, was almost as far as they got. They had (I imagine) no petitionary prayers to God ; but preserved the vestige of thanksgiving. (Those under special Elvish influence might call on the angelic powers for help in immediate peril or fear of evil enemies. ) It later appears that there had been a 'hallow' on Mindolluin, only approachable by the King, where he had anciently offered thanks and praise on behalf of his people; but it had been forgotten. It was re-entered by Aragorn, and there he found a sapling of the White Tree, and replanted it in the Court of the Fountain. It is to be presumed that with the reemergence of the lineal priest kings (of whom Lúthien the Blessed Elf-maiden was a foremother) the worship of God would be renewed, and His Name (or title) be again more often heard. But there would be no temple of the True God while Númenórean influence lasted."

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