In my games, the Cleric is not actively channeling a God's power. The cleric is tapping into the Divine power source. A god acts as mentor to the cleric if they swear oaths to them, but has no power to turn off the connection. Gods and mortal religious entities certainly don't advertise that fact, but clerics who turn away from their patron discover that truth soon enough.
I adopted that logic a long time ago (thanks to Eberron) specifically to not deal with annoying game assumptions of gods turning off the "spigot".