And yet the Romans did have a form of welfare and Jesus did not condemn it or suggest that taxes not be collected to support it.
If government does supply aid to the poor, what would Jesus say to taking that aid away from the poor? If our understanding of Christianity condemns abortion and as Christians we feel compelled morally to pressure government to pass laws prohibiting abortion, then why don't we also extend to government the duty to care for the poor? Why not pass laws requiring farmers to leave a portion of their crops for the poor? Why should one be a matter of choice and the other not?
Apples and oranges. The government we have empowered is charged with those matters we can't individually do. They are not given the power over our lives unless we give them that power through our own lawlessness. They have taken power over the lives of its citizens and it's time its citizens wrested that power away.
If the government is going to provide some great societal safety net, then they are infringing on personal liberty. You are not truly free if you are not free to fail. In the original wording and understanding of the Constitution, the Federal Government was not allowed to deal with us as individuals. That precluded income taxes and welfare until the Constitution was amended. I don't consider this a good thing. We are now enslaved by our government either by the tax yoke or the welfare curse.
The government cannot care for the poor. The best the government can do is force others to care for the poor by taking money and redistributing it. The government does not have any ability to care for the poor on it's own.
We all have a duty to care for the poor that has nothing to do with the government. If we get the government involved, given the nature of government, we are mostly caring for the government, with pennies on the dollar going to the poor.