First off, of course I went with a monotheistic church and a large number of saints and sub-orders within the holy church. There's a fair amount of evidence that EGG also used quasi-Christian churches in early Greyhawk - there are references to the Church of the Blinding Light and the Church of St Cuthbert in Dragon, and one of the only Greyhawk modules with a home base, Village of Hommlet, very much postulates a Christian, medieval setting. Besides, Gary made the cleric class like it was a medieval templar, instead of a wavy-dagger-wielding priest of Baal or Dagon. Or something. Since the Greyhawk pantheon didn't get bolted onto the setting until later, I'm comfortable ignoring it.
There's still a place to have pagan deities - particularly as antagonists. One sensible way to reconcile a monotheistic creator deity with various petty godlings is to make the creator something like the prime mover, the uncreated creator; Perhaps God is something that encompasses the entire universe, like in the philosophy of Spinoza or George Berkeley, or exists outside of creation (or at least outside of this plane). Wherever the petty gods came from, or how they came to be, they are lesser beings and on a completely lower plane of magnitude. (And they can be killed...)
You certainly see this approach in some of the pulp fantasy - I just finished The Broken Sword, where Odin and company want nothing to do with God and what they call the 'White Christ'; Christianity is on a power scale above and beyond them, and they don't want to wake the sleeping giant.
I do have pagan priests in the setting, associated with the fallen gods of the past; they're called DRUIDS. Rather than modify the cleric to be something else, I've loosened up the Druid and made the Druid a catch-all for pagan worship.
I also didn't want to deal with clerical demographics, universal magical healthcare, questions of Raise Dead, or the touchy subject of "do good churches charge poor people for healing". The church is a non-magical institution, and most clergy function like real-world priests - they lead worship, counsel, bless, and act as scholars. Actual spell casting cleric class characters are rare, and chosen.
I was influenced by the Old Testament, where seemingly random guys are selected to be prophets, leaders, and carriers of the divine message; often they're subversive outsiders, or humble folk with low status, not a part of the power structures or the established order. They've been divinely ordained to take a message back to society, to help put things back on course (once, or if, they figure out what is their role). Clerics in Gothic Greyhawk are the same way; they’re chosen for reasons they don't understand, their powers are miracles, and when they show up, they're expected to be setting-breaking game changers - they threaten the social order by their very existence.
Finally - alignment. I'm using the classic rules set, so alignments are Lawful, Neutral, Chaotic. Everyone alive is Neutral, by stint of birth. I'm not interested in monitoring player behavior, so alignment is a supernatural status, and not a measure of ethics or behavior. The alignments of Law and Chaos reflect allegiance to outside powers and are typically beyond what mortals achieve; only mortals with supernatural ties gain an alignment of Lawful or Chaotic. Because the divine order represents Law, all spell-casting clerics are Lawful. Most monsters, which have supernatural ties, are Chaotic.
Demons and Devils could be a bit problematic in the simplified Law vs Chaos scheme if you make Law = Good and Chaos = Evil; I don't do that. Heaven and hell are both part of the divine order; even fallen angels came from the divine world, and are part of creation. (And that's not even broaching the topic that maybe Hell is all part of the plan...). In this scheme, both clerics of God and clerics of the Adversary would be Lawful; but they would act very, very differently.
Chaos represents allegiance to those things outside the divine order. It may not hold up philosophically, (if you define God as coterminous with all of creation), and certainly not theologically, but it's a playable solution for the game.
Heck, the only reason to keep alignment at all, alignment as supernatural allegiance, is so that some of the spells and magic items still work (detect evil actually becomes detect chaos, for instance).
One of the things I love about D&D (as opposed to 1E) is there isn't a whole lot of baggage around the implied cosmology, and it's very much open to this degree of tinkering. May I never again have to determine how to role play a neutral good NPC with chaotic tendencies, or answer questions of whether good characters kill evil monsters when they just want to talk, and whether a certain act violated lawful good, neutral good, or chaotic goodness.