Should Churches Be Tax-Exempt?

If there is separation of church and state, why is it that churches are tax-exempt?  Why is it that the government is giving tax breaks to religious organizations?  

The point behind certain organizations, businesses, and groups being tax-exempt is that they provide a public service.  Usually, these groups engage in activities (such as helping the homeless) that do not bring in much money, and the government wants to encourage them to continue to do their work.  The government does not have an interest in helping social clubs.  

So, where do churches fit in?  What public service do they provide?  

Some would say that churches do help the homeless and provide other public services outside the walls of the building.  This is true, but only a small percentage of the money churches bring in usually goes to these services.  If our goal were to help these services, we would probably do more good by giving to them directly.  Certainly, a larger percentage of our donations would go to helping the people in need it if went to them directly.  

I think the real answer that churches should try to give is that the public service they provide is in the moral improvement of the people who attend.  If they actually make people into better/more moral individuals who (less frequently) steal, commit murders, get divorced, lie, etc., then this would make good sense.  

My question is: Do churches actually do this?  

Suppose you took a random sampling of people and watched what they did during a given day via 24-hour, closed circuit tv.  Outside of watching which ones went to church or said "Christian" things, could you tell which ones were Christians and which ones were not?  Do Christians act any different at work or at home?  When they’re with other people?  By themselves?  

If you notice, I haven’t answered the question "Should churches be tax-exempt?"  The purpose of writing this was not to give an answer but to get us Christians thinking.  What benefit is it that we actually provide?  Are we making a positive difference?  If not, then what are we doing?