The Russian formalist Mikhail Bakhtin wrote about the concept of "Adventure Space" a sort of imaginary space where adventures can happen, which frontiers of all kind definitely fit into (but which are not solely frontiers -Cyberpunk in a way is a dystopian future adventure space). So for Howard, Adventure space was sometimes inflected by Westerns and sometimes by stories of the middle east. For British people of a certain time, Adventure Space would have been the colonies, India or Africa. Space can be an adventure space, often inflected by both of the above kinds of colonial adventure space. (It has often been pointed out how much some space opera stories are structurally westerns to the extent that Firefly went and paid deliberate homage to that idea). One of the thing that defines an adventure space, is not that you can tell one story in it, but that you can imagine a whole lot of stories, en endless amount of stories being told in the same space. This is the space where stories (plural) happen. So of course these kinds of spaces are very important to rpgs (and to pulp fiction where a lot of stories needed to be churned out very quickly.)
In terms of D&D if the western is not the main influence in terms of adventure space, then what is? These things evolve, they don't emerge out of immaculate conception. The Lord of the Rings is one influence, but it is one story, it is the story of one world. And while the Lord of the Rings had lots of imitators, and created in a way a kind of epic fantasy adventure space, that space had a major problem in terms of repeated stories - because the type of story that it was, was the story of the epic decision point of a whole world - so every Tolkien imitator had to make their own world.
So these would seem to me to be the main adventure spaces that influence D&D.
- The Western (In the broadest sense of an imaginative frontier, a not quite settled place that is someone out there)
- The 'oriental' (because orientalist) colonial adventure space (which by the time of D&D was already being forgotten other than by being translated into the Sword and Sorcery Adventure Space.*
- The Lord of the Rings/ Epic fantasy adventure space (which I feel is important but incomplete on it's own).
You can see the process of recasting and reskinning adventure spaces in various works of fiction. Rosemary Sutcliffe wrote adventure stories about Roman Britain and it's fall and aftermath (and apparently a big influence on her was Rudyard Kipling, which should tell you a lot about which adventure space she used to structure her own adventure space). David Gemmell wrote books in a world that was very Sword and Sorcery in detail, but his stories were clearly lacking in the existential themes of sword and sorcery and instead very much focused on the moral themes of westerns. In the 1950s, (or 60s) there was a Robin Hood TV series on American TV that apparently everyone knew was basically a western. Star Trek was sometimes referred to as "Wagon Train to the Stars".
*I feel that this particular influence mostly fell away due the uptake of the epic fantasy aesthetic style which is much more familiar in terms of basic detail (Northern wilderness) and moral themes (Good vs Evil) to most Americans.
Edit: Not all stories have an adventure space or need one. It is not a universal feature. As I said above, Cyberpunk has become a sort of adventure space, where we can imagine an endless series of stories taking place (whether they are good stories or not is immaterial). In contrast, try to imagine a 1984 adventure space. How many stories can you really tell in that world/set-up? (Not that you need more than one).
