there are many fans of Tolkien who believe that almost everything in D&D has antecedents in Tolkien
I don't believe there are
many such fans.
No one thinks that all the Greek-inspired, or mediavel bestiary-inspired, monsters in the MM are from JRRT (eg chimerae, catoblepases, pegasi, hippogriffs, tritons, dryads, basilisks, etc).
No one thinks that chromatic dragons are inspired by JRRT.
No one thinks that all the dungeon monsters (rust monsters, gelationous cubes, black puddings, etc) are inspired by JRRT.
No one thinks that the Girdle of Giant Strength or Hammer of Thunderbolts is inspred by JRRT.
Etc.
The claims about Tolkien influence are very specific. It is particular story elements (ents, balrogs, orcs whose tribes fight one another, rangers who use palintiri, etc); and also the whole notion of fairy tale peoples as naturlised "human" cultures.
if you play OSR/1e/Holmes Basic, you end up with an exceptionally different experience that matches, much more closely, to the S&S model that Gygax preferred.
Conan quite often doesn't get the treasure, sacrificing it for the moral choice instead.
And Conan almost never plans, or manages his adventures in a logistical fashion.
The classic D&D experience - with its emphasis on treasure recovery and logistical management - shares some tropes with REH's Conan, but won't reproduce those stories anymore than it will reproduce LotR.