Not entirely true. Rumil of Tirion is supposed to have written several books, which were continued by Pengolodh of Gondolin. They must have been preserved into the Third Age because Bilbo is supposed to have translated them into Westron.

They were written as historical facts of ME, rather than 'holy' books, but the distinction is a fine one. Rumil's creation story, Ainulindalë, is the opening chapter of the Silmarillion and it is both history and mythology: historical in its account of creation and mythological in its removal from the state of affairs at the end of the Third Age.

It is unclear what the peoples of the TA believed in what we might call a "religious" way. If the Dwarves considered their own tales as fact then Elvish tales would be less significant, but if they considered thir own tales to be parables just for Dwarves, then maybe they could be more accomodating of Elvish tales. How would Elven tales like Ainulindalë be percieved by Men and Hobbits? Wouldn't they be considered as myth? After all, ME was no longer lit by two trees, nor was it flat, so the stories would not match observable phenomena. Even the mountains and rivers had been completely reformed, so there was little possibility of paleaontological/archaeological evidence to validate or re-write the tales. It seems logical to imagine the younger races, as the centuries of the Fourth Age pass, calling Rumil's works "sacred" or "holy", if not to themselves then at least to the Elves.

Indeed you're correct about the King leading the "worship services" on the Meneltarma.
I still think my idea could be valid, though, with the thanksgiving and prayers directly to Eru being not only sanctioned, but perhaps even mandated by the Valar. Why else would Manwë have felt it necessary to send Eagles to "witness" the ritual? He didn't need them to know what was happening: he could see it with divine power. It looks to me as if the Eagles were a reminder to Númenor that they were always being observed to see that their allegiance was properly placed.

The Dwarves reverenced Aulë, so I would think they would see Elven histories as factual historical tales, at least as they concerned the Valar. Anything to do with Elf/Dwarf conflicts, like the events surrounding Thingol's death, would doubtless be read with a jaundiced eye though.

The survivors of Númenor would at least have put stock in the First Age histories, especially since they would have had at least oral, if not written records of some of those times themselves. They also had artifacts like the ring of Barahir to support belief.

Lesser Men like the Rohirrim were probably a lot more ignorant of such remote times, even though they knew of the Vala Oromë.

As for Hobbits, though they held to the Elvish manner of referring to the Sun as she, I doubt they knew the genesis of that. Hobbits had forgotten their own history up to a fairly recent point, so it seems unlikely they knew (or cared to know) about a lot of "Elvish nonsense). ;)

Especially since, prior to Bilbo's work, it's unlikely that the legends/histories of the First Age were even available in Westron, at least in the North; and very very few hobbits ever learned Sindarin. Note that at Weathertop the tale of Beren and Luthien was new to them; and none knew about Gil-Galad save Sam (and doubtless Frodo), who had learned it from Bilbo.