The Cleric has always existed in the D&D game as the third Class (preceded by Fighting Man and Magic-User, and slightly earlier than the Thief). The Cleric role has always centred around being the party Healer, making it vitally important in some games.
However, it’s archetype has never really featured much in fantasy literature,
It no longer matters. D&D is its own canon.
as the number of Classes also expanded I wonder whether it can really be considered more ‘Common’ an adventuring Class than some of the others?
Are you seriously making the assertion that in the last 15 years (since the launch of the original class-glut edition) the Cleric has fallen more out of use than the
Fighter? That the Cleric's universally superior healing skills are somehow less relevant than the Fighter's often negligible combat superiority in comparison to the Barbarian, Paladin, Ranger, and Assassin Rogue?
I question your very thesis, sir. On what are you basing this theory?
Even in 4th Edition, when the concept of Cleric or healer was conflated with the Leader role, the only consensus I heard on the topic was that the Cleric was the only truly effective Leader. By comparison, many of the Defenders who shared that role with the Fighter were far more effective in combat.
What is the relationship between Clerics and Paladins, or Druids for that sake?
Well, in direct response to what I read as your intent, the cleric virtually exclusively casts spells (which can heal injury) and channels divinity (which can heal injury), while the paladin and druid
do other stuff that diminishes their healing capacity, either directly by game balance or indirectly by providing more effective combat options for them to choose.
More importantly, the role of a priest is to administer to his or her flock. In the case of the D&D cleric, that flock is the party. Believers or not, his companions are the cleric's responsibility. The paladin's first responsibility is to smite the infidel and the druid's first responsibility is to the environment. Both classes may have minor ministration responsibilities but it is not the purpose of the class. The ministration of the cleric is part and parcel to D&D lore, regardless of its presence in irrelevant external fiction.