OK. Let's look at my argument. I posit that the core purpose of government is social order. I also imply that this is primarily provided through rule of law (police and legislatures) and sovereignty (military protection).
You state that it is difficult to have social order without welfare and social security, because the poor and elderly would be dying on the streets. This example doesn't dispute my core argument that government provides social order. In fact, it supports it, showing that social order is necessary and provided through government.
Remember, I argue that the core function of government is social order (which you seem to agree with), and that the primary means of providing this order is through police and military. I agree that there are other secondary means of providing social order (welfare and social security), but they can all be linked to the primary means of social order.
Here is an example: Let's say that we do not have any form of welfare or social security (America didn't for nearly half of its existence). When times get particularly tough, the poor may start stealing more in order to survive (notwithstanding the impact of food banks, churches, and other charitable organizations). If theft goes up, the government's primary means of providing social order is strained, because police must protect against more thieves.
Here is how I support my claim that police and military are the primary means of providing social order. Can our government exist without welfare? It has. Can it exist without the public school system? It has. A government that ONLY created and enforced rules is still a government. A government stops being a government if it cannot create and enforce laws, even if it provides all of these other things, will quickly fall into anarchy. Why pay taxes if there is no threat of imprisonment?