There's a name for when the churches ran the world: it's called the dark ages for a reason. I'd rather not have to rely on someone's (supposed) piety on top of the questionable scruples of the church to render assistance. What happens if you're an outspoken critic of the church or (heaven forfend) a "sinner" homosexual? At least when the Tea Party criticizes the government, the government doesn't pull the welfare checks of its members.
They nail them with a tax audit when there's reason to believe they're cheating on their non-profit status. And if they have nothing to hide and their accounting is honest, then an audit shouldn't be a problem. Right?