There'd be some knock-on effects to consider if you did this.
You'd have to rethink how (or even if) the gods would play into the game. Right now, Clerics are a fine conduit to allow deities to mess with things if so desired. Remove Clerics and that goes away. Some might like this. Others might not. Either way, it's a significant change.
Also, at all levels Clerics do something else very well beside just healing, and that's divination: Portent, Augury, Divination, Commune, etc. Where does that go?
ALast but not least: who brings back the dead, or sends them on their way; and who deals with undead?
Lan-"the main problem with Clerics is they are like the bass: easy to play, but astonishingly difficult to play well"-efan
Presumably people would still worship the gods, but it wouldn't just be the clerics that got some benefit from it. If you know the correct rituals, make the correct sacrifices, act the right way, then the god you're speaking to gives you the benefit you're asking for. Very legalistic, most ancient pantheons, in that respect. And if you want your god to pay attention to you, act the way they would and perhaps they'll talk to you in your dreams.