Just because Gygax said that at the time doesn't make it true. The entire core of D&D is based on Tolkien - the mixed race adventuring party. But, that's for another thread.
And, one of the criticisms of Tolkien is that his characters are rather simplistic - the bad guys are 100% bad and the good guys are almost all 100% purely good.
I don't see any reason to doubt Gygax's statement that Tolkien wasn't one of the primary influences on D&D.
"The entire core of D&D is based on Tolkien" ? Examining OD&D shows otherwise:
- The magic is from Vance, with a nod to De Camp & Pratt.
- The Law/Chaos alignment system is from Poul Anderson and Michael Moorcock.
- The illustrations in OD&D, although crude, tend to be more Sword & Sorcery oriented than High Fantasy oriented.
- The forward to OD&D uses John Carter, Conan, and Fafhrd & The Gray Mouser as examples.
- Elves and Dwarves both feature prominently in the work of Poul Anderson. In Three Hearts and Three Lions, one of the main character's adventuring companions is a dwarf. In The Broken Sword, the main character is a changeling and is raised in the land of the elves.
- Elves are central to the book The King of Elfland's Daughter by Lord Dunsany, which is another of Gygax's OD&D influences.
- Michael Moorcock's Elric is essentially an elf, although his race calls themselves "Melniboneans". The OD&D fighting-man/magic-user elf is much closer to Elric than Legolas.
This isn't to say that Tolkien had zero influence on D&D, but the idea that the "entire core is based on Tolkien" is quite clearly factually incorrect.