Maybe Gygax thought it would be more fun if the races were actually diverse, and had different strengths and weaknesses?
Possible (though they don't seem to get much for their trade-offs)! He's pretty clearly on the record as disliking Tolkien, though. He's said he "yawned his way through" the books, and said this about them in regards to D&D:
Tolkien includes a number of heroic figures, but they are not of the "Conan" stamp. They are not larger-than-life swashbucklers who fear neither monster nor magic. His wizards are either ineffectual or else they lurk in their strongholds working magic spells which seem to have little if any effect while their gross and stupid minions bungle their plans for supremacy. Religion with its attendant gods and priests he includes not at all. These considerations, as well as a comparison of the creatures of Tolkien's writings with the models they were drawn from (or with a hypothetical counterpart desirable from a wargame standpoint) were in mind when Chainmail and Dungeons & Dragons were created.
Take several of Tolkien's heroic figures for example. Would a participant in a fantasy game more readily identify with Bard of Dale? Aragorn? Frodo Baggins? or would he rather relate to Conan, Fafhrd, the Grey Mouser, or Elric of Melnibone? The answer seems all too obvious.
edit: I would guess that, regardless of any underlying motivation, it got into early D&D simply because some situation arose where a playtester's elf died and Gygax decided Raise Dead wasn't enough to bring him back for some unknowable reason and it wormed it's way into the final rules. You have to remember that game design for RPGs wasn't the kind of battletested thing it's become today, and AD&D in general was kind of a mix of disparate rules cobbled together from wargames.