As you said, there are three types of religions, and coincidentally there are three classes that get power from supernatural outside forces to represent them: cleric, druid, warlock. Seems doable under the 5e PHB with a few subclasses added for flavor. Unless you are a die-hard who feels clerics and warlocks shouldn't exist or need to be totally rewritten so as not to break the established cannon.
For "typical" cultures, the following arrangement for the three religions can work.
• Elementalism ≈ Cleric
• Animism ≈ Druid
• Divine-king worship ≈ Warlock
There can still be exceptions. Individual Clerics could be any of these three religions. By canon, elementalists or divine-king worshiper. So any of the three seems fine as a possibility.
Warlock can work well with sorcerer-king as the Patron. In this case, the Warlock is an arcane mage, so would be a defiler, which makes sense flavorwise.
Meanwhile, certain features of the Warlock can be explained as the result of earlier magical experiments by the sorcerer-kings, akin to magic item creation. When it comes to the spell slots, however, it makes more sense to me, if the sorcerer-kings know the spells that the Warlock gets from them.
A convenience of the sorcerer-king employing the Warlock class is, it is a popular class, and this flavor can fit seemlessly within the Dark Sun setting.
Then you need a new DS PHB.
DS requires a rigorous rewrite of the PH flavor for the Warlock. If a sorcerer-king as a Patron feature comes with mechanical variants that differ from fey fiend, or goo, then it might as well be a full rewrite of the Warlock class as it pertains to Dark Sun.
It occurs to me, the Athas Elf is unlike the other 5e elves. For example, it seems to me, it associates more strongly with Constitution and Dexterity, and perhaps Strength, as a tireless athletic runner who survives extreme environments. (Compare how the Sea Elf associates Dexterity and Constitution.) The Athas Elf might include psionic features.
In any case, most of the rewrite is to get the Dark Sun flavors right. By the time a writeup mentions mechanical variants, one might as well rewrite the entire Players Handbook entry.
Which is why I go back to a and b options.
Consider the essential 2e Dark Sun getting a full-on rewrite of the Players Handbook.
Full writeups for the essential races/species:
Human, Elf, Half-Elf, Dwarf, Half-Dwarf, Halfling, Half-Giant, and Thri-Kreen.
Full writeups for the essential archetypes:
Champion (Fighter), Battlemaster (Gladiator), Thief, Assassin (Bard)
Cleric, Druid, Warlock (Templar)
Wizard (Preserver, Defiler)
Whatever the Psionic options will be
Finally, there can be a separate section with advice for how to introduce other Player Handbook options, like Paladin, Sorcerer, etcetera, if players and DM want to go this route. But full writeups seem unnecessary for this extra stuff. One concern is to heighten the preserve/defile flavor. So would an arcane Sorcerer also need to choose between preserve and defile? Would a Bard have to choose? Is it better to give the Dark Sun Sorcerer and Bard a psionic tag? And so on, to think carefully about the implications if other 5e options are to be introduced into this setting.