De Camp makes several appearances in Appendix N. I haven't read all of these works, but the ones I have read demonstrate some characteristics I get from Gygax's worldbuilding and from his early fiction that I have seen. There is some content similarity (like universe-hopping, crossovers with the works of other authors, and hierarchical planes of existence), sure, but what I'm mainly thinking of is more of a structural or attitudinal alignment.
For one thing, I think it's fair to say that Gygax's work shows a concern with realism and degree of pedantry around certainly topics: Extensive list of polearms, obscure terminology, etc. De Camp gives us an extensive exegesis of REH's naming in the Conan stories and also an analysis of the same stories' technology. He wrote a series of Sword & Planet stories (the Krishna series) that makes a point of addressing the unrealistic elements of Burroughs' and others' similar stories.
It seems to me there was a logic to Gygax's D&D work. I'm sure this is in part due to it being in a game where you have to be prepared for player action, but it resembles the application of rational consideration of elements in fiction as in the Harold Shea stories or The Carnelian Cube.
Both men also have a fondness for humor in their fantasy. While this isn't an uncommon trait and is found in the work of a number of Appendix N or adjacent authors, I feel like use of anachronism for humorous purpose is something found in Gygax's work that also occurs in the Harold Shea series. Less than totally heroic or unheroic protagonists (often the humorous effect) probably describes a lot of D&D, but also several of de Camp's Krishna novels and his Reluctant King trilogy.
As to Gygax's later work, I've only read a couple of the Gord novels and that was decades ago, but I don't recall them being particularly de Campian. Maybe his sensibilities shifted over time or perhaps they reflect a desire to better compete in the fantasy market that existed in the mid-80s. Still, I think on balance, the similarities are there.
...D&D claims to have a polytheistic religion, but you've got both the politics and the critical Cleric class set up as in the medieval Christian world, and nowhere else.
Is there a more interesting and perhaps more realistic way weave together the elements presented? I think so.
Note that Clerics of 7th level and greater are either "Law" or "Chaos", and there is a sharp distinction between them.OD&D mentions Law and Chaos with regard to a cleric's orientation. To me, this suggests a system of belief with a dualist cosmology. (Perhaps this is the actual state of the cosmos, but it doesn't have to be!) This is a moral dualism, as the two opposing forces or principles are in conflict. This could be interpreted (and perhaps is by some sects or particular faiths) as ditheistic with two gods or groups of gods in opposition, but I also think the broader, philosophical tradition could embrace transtheism, where the existence of Law and Chaos is a greater and more important truth than the existence or nonexistence of god-like beings/powers.
The church of law is syncretic, incorporating deities as it grows as agents, exemplars, or aspects of Law. No doubt there would be historic disagreement (possibly even conflict) over just how much deference and attention these powers are rightly due.
Clerics/priests, given the hierarchical structure presented in OD&D, are important in public rituals and ceremonies of the belief system but are also likely interpreters and scholarly experts on Law. Each of these Patriarchs (and Matriarchs, probably, though OD&D doesn't mention them!) is independent and self-governing but in fellowship with the others (generally). Initially a Patriarch would be a charismatic leader who attracts followers, but presumably the church they founded would have a mechanism of choosing a successor.
Patriarchs are the final arbiters of the commandments of Law within their area, but the Patriarchs of the various churches might vote to decide points between them, or perhaps different interpretations would reign in different jurisdictions. Another aspect of the high clerical function extremely relevant to adventuring is calling for and supporting crusades/jihad against Chaos.
Speaking of Chaos, it does seem a bit odd it is presented with a hierarchical clerical structure identical to Law's. One possibility is the "anti-clerics" are sort of Satanists and just performatively mock the church of Law, but another possibility is that "Chaos" only speaks to its ultimate goals or cosmological beliefs, not to its organizing principles. It's also possible (even likely) that the Church of Law applies the name Chaos to a diverse group of belief systems that don't agree with it and often don't agree with each other.