It has become WAY more than a safety net. I'd call it more of a safety web because when it catches many people they are stuck in it.
I agree that it doesn't work well as currently designed. I believe that we need to help out those who are in dire need of help — as a prosperous society we should not ignore those facing hunger, homelessness, etc through no fault of their own. Indeed, Christianity teaches giving to the less fortunate as central tenet of the faith. So I would say we ought to offer help when it's needed. But that help should be effective in both results and cost. Our rather disjointed set of federal and state aid programs today don't meet either test. They are not particularly effective at lifting people out of poverty or very cost effective at what they do. There is a lot of room for improvement here.