I think you missed the point.
Of course I know that many...even most...monsters in D&D have sources other than Tolkien.
My question is whether we would even have "fantasy roleplaying" (or fantasy as a genre) in the absence of Tolkien. I am not suggesting he is the source of every idea, and yes he in turn got many of his ideas from other sources including English and northern European mythology (I'm surprised nobody has mentioned William Morris yet.)
But he popularized the genre. I don't think it would have occurred to the wargamers of the 70's to create fantasy roleplaying if Tolkien had died in WWI.
Yes, you are absolutely correct. Tolkien was hugely popular in the 1970s. The popularity of Lord of the Rings brought a lot of older fantasy to light. New authors began publishing fantasy, which wouldn't have been financially feasible for publishers without the professor's work. Science fiction authors and fans were aghast, because fantasy was eating into limited shelf space, and to make matters worse, a lot of it was very badly written. I remember the time period, and how hard it was to find the jewels amongst the crap.
In this climate, adding a chapter on fantasy war gaming for Chainmail would have seemed like a great idea. Gygax would have been appealing to the Tolkien fans, because it was so loved. Of course he drew from Tolkien, even if he himself preferred Swords and Sorcery. He could add elves, dwarves, wizards, trolls and ents, and players would understand the context. Purist wargamers, on the other hand, were very dismissive of this non-historical flight of fantasy.
When DnD was being developed from Chainmail, the Tolkien elements got stronger. Were there other influences? Of course. Did Gygax copy Tolkien faithfully? Of course not. Inspiration was heavily coming off Tolkien, however. Without that love for Tolkien in the 70s, fantasy would have remained obscure. Gygax and Arneson wouldn't have bothered with their fantasy supplement to Chainmail, and therefore, no DnD.
Role playing was an idea that had been simmering elsewhere. It's very possible that other creators might have come up with something. Would it have caught on? Maybe eventually, with a different set of circumstances making fantasy popular. Maybe even a different genre could have got the ball rolling. In the 70s, however, Tolkien popularized fantasy, allowing Gygax and Arneson to create the first commercial role playing game, which caught the public's fancy. And remember, most of the early adopters were university students. University was one of the places where Tolkien was especially appreciated. It all ties together, and yes, Tolkien is a major ingredient.