Yaarel (he) wants the rules to have nothing to do with gods. At the same time, DMs can use whatever campaign setting they find interesting. Some will be monotheistic, some will be polytheistic, some will be other. Some will mix it up, like Eberron.
I'm sorry, but, can you be a bit more specific here. What rules are you specifically talking about. The mechanics for Clerics don't reference polytheism or monotheism at all. While the flavor text does, the mechanics don't. You can easily represent different traditions within a monotheistic setting by using the different domains. So, a Knights Templar organization would be War Domain Clerics, while, Knights Hospitalar would use Life Domain, while some sort of Church Archivist organization would use the Knowledge Domain.
((Note, my history here is WAY shakey. Please don't get too hung up on the specific examples, it's not something I've researched, just something I've pulled out of my petoot in the five minutes I've been writing this))
I'm not sure what the issue here is.
It is very difficult for me to ignore the rules as written, when the flavor and mechanics and cosmology, endlessly refer to it.
Now, I'll agree that the flavor and certainly the cosmology is going to be an issue. I'm actually right behind you on this one. I think 5e has gone too far in hard wiring too much flavor into the game, but, frankly, that ship has sailed. People wanted the flavor and well, it sucks being on the wrong end of the stick.
But, in any edition of D&D, you'd have to eject the cosmology in order to do a monotheistic campaign. My advice would be to go the Primeval Thule or Dragonlance way. Simply pare down the cosmology to what you want to use - and don't use planar travel in your campaign or, rather, don't use it much.
For myself, I cannot use rules as written that contradict the narrative immersion in the campaign setting.
This is a lot more difficult to unpack. At some point, it might be better to simply admit that D&D is not a good fit for a particular setting. Where that point is will vary from person to person. But, again, can you be specific about the mechanics that contradict the narrative immersion in the setting? Flavor text is going to have to be rewritten in any campaign setting. That's just the nature of the beast. But, what specific mechanics are causing problems?
The Cleric class comes with too much setting-assumptions baggage.
The Class would be more useful its rules and descriptions were more open to the various kinds of settings, and various kinds of religious traditions that might happen in those settings.
Alternatively, maybe the Cleric is a campaign-specific class, like the way the Artificer class is specific to Eberron. If so, the Cleric is a specialty class for the Forgotten Realms/Planescape setting (5e seems to have merged them into the same setting). But if so, then the Cleric is no longer a core class. Make sure that Druid and Bard have optional builds that can heal equally well as the Cleric. Maybe even the Paladin can have an Oath that is a premiere healer, including all the healing spells True Resurrection at the high level, with bonuses to healing.
Really? Now, let's back up a second. The Cleric, as written in D&D, is a purely D&D construct. It really has no genre equivalent. And it never really did. It's not so much that clerics are setting specific, but, rather, D&D specific.
Now, that being said, I look at how clerics are detailed in Primeval Thule. Granted PT is a polytheistic setting, but, it ejects much of the standard D&D cleric. Clerics in PT are not granted spells by their respective deity. They learn spell casting from other clerics and are effectively just another kind of wizard with better organizational skills. Churches are more cabalistic cults focused on specific individuals and there are no behavior ties between your cleric and his or her deity. If you want to worship a good deity and still be an evil bastard, you most certainly can. And vice versa.
So, I'm not really seeing the issue here. It's been done and it's been done in 5e. I would think that a monotheistic setting would be a heck of a lot easier to design than a polytheistic one. You only have to detail one deity after all.