Yes and no.
The problem with the government is that it is an institution and not an individual. Institutions don't care. Individuals do care. Individuals within an institution can only act on their impulse to care to the extent that the institution allows. So in that sense, the original meme is correct.
However, the person who made that original meme is discounting the fact that a democratic government is an institution that is moldable by the will of voters and their elected officials. If the voters cared more and elected politicians who cared more, the government would react more positively to the people's needs.
The problem is when you have a population so jaded and pessimistic that they don't think it's even worth trying to make government respond more productively. They think the best they can do is make it so that government has as little influence in their life as possible, as to minimize the damage that government might inflict on them personally. These are the individuals who make up the Republican voting base, and they are also the main reason why the government doesn't do more to help people.
It's a self-fulfilling prophecy: I don't trust the government because it's broken, so I'm breaking the government to ensure its brokenness doesn't affect me... And when the government isn't there when I need it? Because it's too broken to do its job? It just proves I was right all along.