Would they help?

I've been trying to keep up with all of the budget talks and keep informed as much as my average sized brain can handle.  But I read an article about a lobbyist who believes in: "small government, lower taxes and limited government services." 
Ok, there needs to be support services, and churches (all religions) really did pick up the slack for the poor in the communities in days of yore. The rich donated to churches and then they in-turn filled that niche. Then slowly government started picking up the tab for those services.  I guess, and I am making an assumption, taxes went up and donations to churches went down. Non-profits sprung up to help fill the holes, taking the place of churches since they couldn't do it anymore.  Non-profits were very specialized to meet a specific need. Basically, money went from one organization to another to pay for these services. It's all a game of shifting money.
Now, if we reduce government assistance for the needy and poor, and cut taxes (specifically for the rich) will the rich go back to donating (and I don't care if it is to a non-profit or a church), so that those services can be provided? Will we as a society go back to helping our fellow man primarily through churches and non-profits, instead of paying the government to do it?

And, playing devils advocate, are non-profits and churches better equipped to decrease fraud in providing help for the needy and poor than the government is?  I don't necessarily believe in big government or small government.  I like the idea that whomever is more equipped to handle the task with as little corruption as possible should do it.

Things I think about.