Consider the following article on Gygax's early play with fantasy miniature rules. Note how the entire context is inspired by, and driven in terms of, representing figures mentioned in Lord of the Rings:
(There are places where Gygax considers some alternative statistics to be better than Tolkien's, but nontheless, the idea for what character to begin playing with in the first place comes directly from LOTR.)
More keenly, consider the following list of figures presented in the Chainmail Fantasy Supplement. Note first that it begins with hobbits (!). Secondly, it seems like a pretty clear line can be drawn from the LOTR books directly to this ordered list:
Halflings, sprites (& pixies), dwarves (& gnomes), goblins (& kobolds), elves (& faeries), orcs, heroes (& anti-heroes), super-heroes, wizards, lycanthropes (shape-changers), trolls (& ogres), giants, treants, dragons, rocs (& wyverns & griffons), elementals (& djinn & efreet), basilisk (cockatrice), chimerea, giants spiders and insects, giant wolves, wights (& ghouls).
(Question: Of these 21 distinct creature type categories, how many are not found in the works of Tolkien? Any?)
Of course, these basic types are generally the same types found in the two-page listing of monsters in OD&D. I would agree that starting with Sup-I Greyhawk, the monster and spell lists are being created uniquely for the D&D game, but not before then. It's a bit silly for Gygax to argue that "some bits and pieces from Tolkien were thrown in to entice certain readers", when the fact is that all of the monsters and character-types of Chainmail and core OD&D come directly from Tolkien, nearly in the same order of appearance as in the LOTR books. Tolkien is the first fantasy author mentioned in Chainmail fantasy, and Gygax's earliest writings show that the whole game was played in that context.
Hey, that's not a bad thing! Tolkien is great, strong foundation to build from. I can see the Tolkien-to-D&D relationship as being analogous to that of Games Workshop-to-Blizzard corporation. Our art is strongest when there's lots of borrowing going on. Fie on silly copyright regimes that restrict and traumatize even our greatest creators to such a degree.