Yeah, the timeline doesn't match up, it seems. Actually a good third of this essay is Anderson comparing his approach to Elves with Tolkien, and what different influences and goals they were going for in their stories. A very good analysis, really, Anderson was a smart dude.
That said, I firmly believe that Gygax was more influenced by the Hobvit than he wanted to admit after being sued.
On that .... I think that a bunch of different things can be true. And I will list them:
1. Gygax's statements were often self-serving. I've done enough historical research to know that he would contradict himself, and that he would outright lie about things for his benefit (look at his taking credit for the creation of the Thief class, for example, despite
published information to the contrary). So yes, take his comments with a giant grain of salt.
2. I agree with Jon Peterson's analysis that the fantasy rules for
Chainmail was "based off of" (took large elements from!)
Batle of Pelennor Fields, which was a wargame that was ... yeah, completely Tolkien. That means that the fantasy rules for
Chainmail borrowed from a Tolkien game. And if you look at those rules, you see it.
3. That said, I also completely believe Gygax when he says that he wasn't a huge Tolkien fan. Again, you have to remember his age and his time. For him, Tolkien was almost "uncool," because it's what the "new kids" and the "young kids" were into. The real fans (like him!) were into the swords and sorcery. I believe that he really preferred the multitude of authors he listed, and that they were that influential. It's why he immediately brought Leiber into the fold. Hero worship, of a kind.
4. On the other hand, Tolkien was immensely popular in the 70s, and Gygax was a businessman. And (second-order design!) the people that were playing were also Tolkien fans. So it's no surprise that a fan, for example, wrote in their submission for a Ranger class that later got codified into AD&D.
5. So I think that Tolkien was both more influential than Gygax admitted, but also less influential than people today think. Because Tolkien is so omnipresent, it is easy to both see the things that were obviously "Tolkien" but also to read Tolkien influence into things that were not a product of Tolkien, but actually came from other sources. Most people today lack a familiarity with the other sources and the general gestalt of the time, so everything becomes Tolkien.