If clerics can cast cure disease, why is there disease in your game world? Wouldn't all diseases eventually be eradicated? Wouldn't all those tables about diseases in the Dungeon Masters Guide be rendered useless except in areas where there are no clerics?
Assuming that there are Lawful clerics in your campaign world, why are they not constantly surrounded by ailing people seeking their healing powers? The clerics are peripatetic, you say? They take to the road to spread their faith? That wouldn't stop them from being mobbed by the sick wherever they go. That wouldn't stop the mobile sick from following them, nor would it prevent the newly cured and saved from tagging along. Imagine your small party of adventurers a thief, a fighter and henchman, a magic-user, two hirelings, a cleric, and several hundred of the faithful in various states of health embarking on a journey to raid the tomb of the lich duke. It will get very crowded down there, and you can forget about stealth. The conscience of a truly Lawful cleric would not permit him or her to leave a community where there are people in need of healing. So much for the adventuring cleric.
Well, the D&D cleric is more of a spiritual warrior who is needed to defeat great evil, you say. That explains the cleric's assignment, but it doesn't explain how he or she will avoid being constantly surrounded by the sick and injured unless the cleric's identity is kept secret. Could a cleric be sent on a mission in disguise? It's possible, but unlikely to work. How can a moral healer turn away a wounded friend or even a stranger in need? As soon as someone gets stuck with an arrow it will be cure light wounds here and neutralize poison there. There will be miracles all over the place, and miracles have a way of grabbing the public's attention.
What exactly is the rationale behind a class of individuals who routinely work miracles? At least in Empire of the Petal Throne it made sense: a secretive and jealous priestly caste that had few humanitarian instincts. In Dungeons & Dragons, however, it is much less clear. The cleric class is based on the Christian priesthood, more or less, apparently with elements of Bram Stoker's (or Peter Cushing's) Van Helsing. Imagine what the world would have been like if every medieval priest had been a spellcaster. Imagine what a typical fantasy world in fiction would be like if every priest were a spellcaster. It would certainly alter the balance... of everything.
This is not a rant about whether the cleric class ought to exist. It's a proposal that perhaps it ought to exist differently. For my own campaigns, I'm taking a cue from Roger the GS inspired by FrDave and replacing cleric with prophet. Prophets are not nearly as numerous. They are not necessarily members of the clergy and their talents are not necessarily as well advertised. Being very scarce, there is no institution surrounding them and no expectation of how they will react when encountered. In fact, they are usually not even identifiable as prophets unless they introduce themselves. My own prophet class will have a number of other differences, but just the change of name and what it entails has a profound impact on the game world.
Incidently, I revisited this conundrum when a new player heard me mention a "cleric" and thought I meant a secretary. That's another benefit of the name change.