I'd argue that the decline of Conan's relevancy is related not just to his age, or the degree to which he is or isn't read, but to the degree that the "barbarian" figure--the figure free from the constraints of civilization, who relies on extreme athleticism, cunning/craftiness, and wild daring to overcome problems soft civilized folk cannot--isn't a very strong heroic fantasy anymore. That's not to say that this figure is entirely gone from pop culture (I mean, isn't Jack Reacher basically Conan?), but the fantasy just doesn't hold the power it used to.
When I was growing up in the 80s, my fantasies of heroism encompassed figures like Robin Hood, Luke Skywalker, D'Artagnan, and Aragorn. As I became a teenager, with a teenager's desire for things that are edgy and bada**, it shifted to include figures like Blade and Sanjuro. On that list, only Sanjuro comes close to being a "barbarian," but he's not really uncivilized, just disreputable. His heroic value isn't that he rejects the skills of civilization for something wild and elemental, but that he's even better at them then reputable samurai. I could understand the appeal of Conan, but he never really spoke to me at all personally. My daughter, on the other hand, just finds him baffling. They can't even see what the appeal is in the first place.
To me, this seems to be to be related to the way the world has changed in the last hundred years. When REH was writing Conan stories, he was reacting against the interconnectedness of the world--already in full swing both politically and technologically--and his fantasy was a rejection of that "civilizing" interconnectedness. But we've taken that interconnectedness far beyond anything REH could possibly have imagined, and created a civilization that is fundamentally antithetical to the entire concept of Conan. If you've grown up with the internet, Conan's just hard to get.
The way REH has continued to influence the world of fantasy fiction has, I think, less to do with the Conan stories themselves then with the history of "weird fiction." I'm kind of surprised nobody's brought up China Miéville or other "New Weird" authors as the descendants of REH. Where Gygax had REH and Jack Vance, I have Tamsyn Muir and Max Gladstone. Conan may be a shadow of his former self, but things are just as Weird up in here as they ever were.