No, in a capitalist system government should not be responsible for funding and providing all of those things. There are two things mentioned in this question that should undoubtedly be the responsibility of the government alone, or almost alone. There is one that the government should not be involved in at all. The others are debatable.
The government should clearly be the most important source of funding for education and public safety. It is not feasible to have a lot of private police forces operating in the community. It might be possible if we all live in gated developments, but until this happens, it will be necessary for government to take the leading role in providing public safety. Education should be provided by the government since it is essential for the long-term economic security of our country.
By contrast, governments should not be involved in providing job security. If governments make it impossible for people to lose their jobs, bad results happen. This is one of the problems that many European countries have. Firms do not want to hire because they will be unable to lay people off if they need to.
As for the rest, these are debatable. Healthcare, housing, and elder care should be a joint responsibility. Not all people should get these from the government, but it is likely that some people should.
Whenever it comes to the point of Government and the citizens -- it is always debatable.
And so long discussion may not help to solve the problems really.
But as an individual philosophy- in a broad sense - I believe - The Government should do all what they can do truly and the citizens should co-operate the government as much as they can.
The bond of Trust between the government and the citizen is I feel very necessary.
I feel - trust is the foundation of any relationship (whether it is between God and Worshipper. Parent and child, King and his people, Or Government and his citizen ) and rest of the things including Responsibility are parts of it.