Now, it's possible that all adventuring clerics have a sinecure in the original sense -- "without cure." That is, a priest serving in a capacity that does not involve ministering to the faithful, with the sacraments. Obviously, this could be a very hard job, not the current sense.
Paganism laid more emphasis on the proper performance on the rituals, especially the sacrifices, than D&D generally prescribes, but the principle was the same. Counseling a poor woman how to invoke the goddess's wrath on the thief who stole from her, using a lead tablet, is not in the normal job description of a D&D cleric. (In Greek paganism, a priest would not only be a priest of a specific god but of a specific sanctuary. Being a priest of Apollo on Delos did not transfer to any position at Delphi. That would be even more ill-fitting in D&D.)
BUT -- even granting the sinecure -- a cleric doesn't have to adventure with reference to the church. At the very least, some of the gold should make it back to the church. More likely a cleric would be sent on a specified if perhaps open-ended task: a cleric of the domain of travel sent clear out the dungeon that's menacing travelers, escort a caravan through dangerous territory, guard a waystation, hunt down the necromancer who is preying on travelers because they are strangers to the land. . . or level-grinding. Level-grinding would cover a multiple of sins by way of excuses.
Still, in the end, the religious structure would expect the cleric to put it to use for their sake.
The role would be much more plausibly described as a devotee. To be sure, one would expect to see some DEVOTION required. But it would explain a certain free-wheeling aspect to the role. And nothing would prevent a devotee from being connected to a church.