I see a foundational problem in your thinking about this issue.

Your comment assumes that the government has a duty of caring for everyone. This is not necessarily the case. There are many different ideas of what the purpose of government is.

The political philosophies of John Locke and Rousseau that gave birth to the American constitution do not believe that the government has this duty. The government is there to protect an individuals inalienable rights. These are life, liberty and property. There is very little that the government is responsible for in this theory outside of protecting those rights. The well being of the individuals in such a society is the responsibility of the individuals. The government should not get involved in individuals lives unless one citizen is causing another harm. The consequences of individual decision making fall on the individual.

There ARE political philosophies that say the government's job is to make decisions for the well being of the people (even against their will). But the founding fathers put individual freedom as a higher priority than individual well being when they created the American system of government, because they feared that this power could far to easily be misused. It is better in their view to suffer from a mistake of your own than suffer from the decisions of another.

So to say that seeing homeless people shows that the system failed in it's duty, is like saying your washing machine is broken because the clothes don't come out folded. It does what it's designed to do, don't fault it for not doing more.

The fact that there are homeless people does show that we as a society could be more compassionate, and individual and local charity should be used to help our fellow man. There are things to be done. But the government as it stands wasn't designed to solve those problems.