The lack of "divine magic" in the pulp tradition that inspired D&D is what makes me wish we could just have one spellcasting class without this weird divide.
Playing D&D with a cleric is like forcing Conan to team up with the Pope to go on an adventure. Spoils the vibe.
You bring up Conan, but that's actually a pretty good example - Howard had a good god in that named Mitra (probably based on the real world one) that battled the forces of another, evil god, Set (ditto)
Mitra is mostly just mentioned, but in one story a queen prays to him and Mitra directs her to Conan. In another, priests used magical powers to hide him. (I think it was Hour of the Dragon)
Beyond that, read some of Manly Wade Wellman's pulp tales (he's listed in Appendix N). The Silver John stuff in particular, but another series featured a former Civil War soldier turned preacher who fought the supernatural.
A. Merritt's stuff also featured priests/priestesses of gods - like
A Ship of Ishtar. And then there was CAS's Avereoigne (sp?) tales, which sometimes have priests confront evil with varying degrees of success - an Abott in one tale chases off a Lamia, while in another, a necromancer handily beats up two novices.
Not to mention D&D isn't just inspired by the pulps (or modern fiction) - it owes a lot to folklore, as mentioned, the Arthurian stuff, but also the later Charlemagne tales (where the Paladin comes from)