Every change made to cleric since OD&D has been to add more specificity in the form of abilities themed to a patron diety or focus of worship, and to attach them the central core of patching up hit points and status ailments.
I think the D&D cleric has an identity as a support character - healing, buffing, divining, usually heavy armor and a shield but not often heavy weapons. Currently, that identity informs the cleric chassis independent of domain - clerics of life, clerics of trickery, clerics of death, clerics of war, clerics of light, all have access to solid defenses, and healing, buffing, and divining ability. They protect and enhance their party members.
The game has made many strides in making other classes able to heal, and 5E is arguably the most...uh...liberal in that regard. As a result, that core identity is softened and so Domains pick up that slack.
If a player/DM/table want to add more specificity to the Cleric by creating specialized spell lists for each Domain I don't see that as a bridge too far. A Paladin, A Cleric of War, A Cleric of War with a specialized spell list, and a Fighter with the Acolyte background and the Magic Initiate (cleric) feat are all very similar, but they are also different enough not to worry about it.
If instead we renamed the Cleric the "Healer" class, made sure the domains/spell lists were different ways to buff/heal that would be better, IMHO. For example: War Clerics would heal and buff with attacks (splash heals, warlord-ish like abilities), Life Clerics would be the ultimate healer, Nature Clerics would heal with poultices and alchemy, Death clerics would heal by "life-tapping" enemies, for example. and so on.
That is, clerics would be better served as healers with different expressions of healing.