How did specialty priests work? I haven't played AD&D in almost 20 years.
That's actually a confusing question. In 1e AD&D, there were not specialty priests, all Clerics were the same. In 2e, there were two different systems. The one I'm familiar with was in the Complete Priests' Handbook by Aaron Alston. In that system, the DM designed specific priesthoods that could be based on the veneration of a Deity, a Philosophy, or a Force (which, ultimately, made little difference). The key design points were combat ability, granted powers, and spheres. The weaker the combat abilities of the priesthood, the more Major & minor accesses you had to Spheres (thematically related sub-lists of the Cleric+Druid spell lists), and the better the granted powers you could get. Turn Undead was an example of a granted power. You could approximate the PH Cleric and Druid using the system, though the PH versions were more powerful (particularly the Cleric).
I think the other system was in Legends & Lore, but I never adopted it, and just remember that it could be decidedly overpowered in some cases.
Since then, Clerics have chosen Domains: Two Domains in 3.x granting fairly 1 extra slot/spell level and adding a special ability and 1 spell/level each to the Cleric's list, optionally one or more Domain feats in 4e, one very defining Domain choice in Essentials and 5e.
To try to relate that back to the thread topic: in 5e CoDzilla doesn't step on the Fighter's best-at-fighting toes so much as in 3.x nor even 1e AD&D. Not all clerics get the fighter's heavier armor, they don't get any extra attacks per action, can't crib the equivalent of fighting styles, and don't self-buff as outrageously. (A Human Cleric can still get heavy armor, snag a feat, and maybe show up a fighter at first level, a little, but it doesn't last past Apprentice Tier, when the classes sort themselves out, and the Cleric becomes all about casting, and the fighter all about weapon-based single-target multi-attack DPR.)