No one in the late Third Age doubted the existence of Elves
This might seem like an obvious point, but it is less obvious when you think about it.
In previous posts, I asked whether Frodo had ever met an elf before Gildor, and whether people would instantly recognize elves as elves.
In the late Third Age, most Men, Hobbits and Dwarves would only have rare chances to see and communicate with Elves, and that was only when they wanted to be seen. And yet nowhere in The Lord of the Rings do we hear anyone disbelieving that Elves exist. Which seems unusual--- immortal beings with strong powers and what seems like "magic", who live in a few enclaves hundreds of miles away, and yet everyone in Middle Earth knows they are real. And in some cases, treat them as a normal, prosaic part of life. The source of bottle deposits, for example!
Even when people react against Elves, such as when the people of Rohan and Gondor show fear or distrust of Lorien, they don't disbelieve that they exist. Despite having what seems like hundreds of years without contact, they just take it for granted that there are kingdoms of immortals living a few day's ride away.
(If anyone can think of any quotes or passages that contradict this, feel free to let me know)
(Also, I know some of this is external to the writing...if Tolkien had planned The Hobbit as part of the larger mythology, he might not have had elves doing something as common as floating barrels down a river to a city of men).