The reason that the government should take care of the poor and the sick is not because it is the most efficient and moral way to do so. Indeed, given appropriate conditions, I concede that "the market" and the goodwill of "the people" would be capable of ensuring that their needs are met in a more efficient and, according to libertarian logic, more ethical manner.
The reason that the government should provide this welfare service, however, is that it results in an incredibly desirable situation where any citizen of the United States would know that, no matter what the state of the (semi-)unregulated economy, it is guaranteed that they will be taken care of.
Were welfare left to the emergent logic of the market, somebody who, say, depends on medication to fight a chronic condition or somebody who cannot for some reason hold a job reliably (like my pan-handler friend who can't get a job because of his face tattoo and rotting teeth) would not be able to count on being able to survive should there be a severe economic depression or a war.
To live in a society where somebody who depends on an external institution to them to keep them alive cannot absolutely depend on that regardless of the economic climate is, in my opinion, uncivilized, and for such a person, downright hellish.
If you could convince me that the market is more reliable over time than the US government, then by all means, I might consider the dismantling of the welfare state a viable option.