If we went by that standard alone, we must logically conclude that babies should be starved, as should people who are for any reason physically or mentally incapacitated. After all, they can't work ... why should they eat?
That's the problem with bumper-sticker morality. It doesn't leave room for rational thought, much less common sense.
Based on government intervention, they should not. Private charity should handle these cases. This method adds the extra feature of being able to appeal to the government if there are abuses in the charity.
If the government becomes a charity, who then can we turn to in order to stop abuses?