I think you have it backwards actually. Safety nets not only catch people who fall, they also encourage people to climb. Without this assurance, would people take as many chances? Getting educations, for instance, moving to new places, or trying to start businesses? Would corporations take as many risks in expanding or developing new products?
Either way it seems silly to blame welfare for anything. It is merely the attempted solution to the real problem, which is a lack of opportunity. Wages have been flat for decades, unemployment is high, worker protections are eroding, and, statistically, socially mobility in America is near the bottom of the pack of industrialized nations.
Until we correct the lack of opportunity and actually start giving people living wages, we will have to care for our neighbors who need our help. That is also part of our great spirit that is constantly and willingly being eroded by people who go around casting disparaging aspersions at people who receive government assistance. You hear that kind of crap all the time and frankly I find it disgusting. These notions of people essentially being lazy losers is complete rubbish and I think people should be ashamed to treat their neighbors that way.
That doesn't mean the programs are perfect, but I think the theory of them is sound. I don't see them being run by the government as being a problem. The requirements should be that people need help.
An alternative argument for the squelching of the American spirit is that it is a result of corporate coercion. Their close ties to the political process, favorable government treatment, tax breaks, monopolies, etc. Can a person, today, working in their garage start a company that will eventually become one of the largest companies in the world the way that Steve Jobs did? I doubt it, and that is unfortunate.