What little he describes is enough to establish that his stories cannot be set during the Neolithic.I'm not sure what "cultural, societal, and technological levels" you're talking about. Tolkien, as an author, describes very little about these things, so I'd appreciate more specificity about what you think is being represented. This is complicated by the fact that Tolkien's fantasy depicts different peoples at different "levels". The Elves, especially those who came from Valinor, have a very high level of culture influenced by divine tutelage. The men of Gondor have knowledge of engineering and other mental abilities that far exceed those of common men. The activities of these more advanced people's were mainly taking place in areas which are now underwater. Tolkien says very little about the cultural realities of the common folk of Middle-earth with which a valid apples-to-apples comparison could be made with the archaeological record. Then there's the problem that what's "attested" can't possibly be the sum total of everything that has ever existed. There has to be some room for imagination, especially in a fantasy where that's kind of the point.
As it has been repeatedly pointed out, metal tools and weapons are common in the stories. The titular rings, Narsil, Frodo's mithril shirt, the Red Arrow, gold and silver in Smaug's horde just to name a few. By definition, this makes the story not-Neolithic, because, again Neolithic is defined as that stage of civilization in which metalworking has yet to be developed.
Writing. Earliest forms of writing emerged during the Bronze Age, at least in Europe and Asia. Technically, in the Hobbit JRRT stiles himself as the editor of Bilbo's diary (There and back again), which was expanded by Frodo and Sam into the Red Book of the Westmarch. Not to mention that books did not assume a form similar to Bilbo's diary (red-leather bound volume) for several thousand years after invention of writing. We could even make the point that Neolithic being a part of prehistory, and history being defined by the invention of writing, this too makes LotR by definition not-Neolithic.
Horses. Current evidence points to horse domestication happening around 3000 BCE (at least as mounts, likely earlier as livestock), so again Bronze Age. Stirrups came much later.
Fireworks!!
Also, note that in the passage you quoted, Tolkien doesn't say that LotR is Neolithic fantasy, it says that he imagined it to be set around 6000-4000 BCE (I agree that "our day" is open to interpretation). There is a subtle difference here, because, again the Neolithic time range in a given area is defined by the technological and cultural advances of the people living there.
I'm not familiar with that letter, but based on that fragment, IMO Tolkien is simply stating a timeframe for when these events happened in his fantasy version of Earth's history, but he is not making any inference on the living conditions of the common folks of Middle-Earth.