I think this is a great question when you consider the points of view of anarchists that oppose civilization and institutions: currently we depend on large social structures to take care of people who may not be able to take care of themselves, for example, the elderly, orphans, and sick. If I were to ask your question, I would remove “the poor”, because poverty is a social condition, and poor people can sometimes take better care of themselves than people who aren't because they are forced to endure greater amounts of stress and obedience than those who are better off and can pay for labor and help from other people.
The ultimate problem with the medical system and other institutions that help those in need is the fact that they are impersonal entities and address certain symptomatic problems, while ignoring everything else. For all the miraculous things the medical system can do for people, it often pays little attention to how modern medicine, surgery, medical technology, and the bureaucratic and slavish jobs/positions effects the quality of life for all human and non-human beings. From having a lot of experience with the medical system at this point, I see a correlation between the ways of thinking in medical science to the ways of thinking in the “pro-life”, anti-abortion movements. “Life is a joyous gift even if it is miserable...” I am not saying that the medical system doesn't have its good qualities and improves the life of many, but to me it's unfortunate that “life” is hallowed out in the sense that people can't see that a miserable life isn't worth living.
Let me give you another example of why our social system only appears to take care of the weak and disabled: consider how life is for elderly people in a social system such as ours. As has been pointed out before, they aren't seen as very “useful” to people earning an income with a job, and when they lose their ability to take care of themselves, they end up in assisted living and nursing facilities and the state takes possession of their house. Having personal experience with this, they end up just being ignored by their families when they are committed, because their basic needs are being taken care of and the point of view usually is “that's all that matters”. But how would you feel if you couldn't get around by yourself, go the bathroom by yourself, feed yourself, do any of the activities you did when you were younger, and then be ignored by your kin while the people who made you what you are are long dead? You are left with a pretty empty existence if you can't start new relationships with the people in the facility: often the only thing people in their old age care about is having social interaction, and more often than not they end up being completely alienated and are stuck with their own minds and a television set.
There are people who have in a crass manner talked about the good of “population reduction” (often very wealthy people) and anti-civ people who have callously talked about how they don't give a shit what the negative consequences of societal collapse are because the greater good of the destruction of civilization is what's important, and this is unfortunately what people imagine when we start talking about when we say society is rotten to the core, it's all hopeless, ect. It's also true that reverting to less technological solutions to the problems of life can also cause more environmental destruction if applied to the massive human population (im thinking in this specific instance, how grid-tied fossil fuels are environmentally destructive, but if people on a mass scale just started burning fires again in really cold seasons/climates then that would be even worse because of all the noxious gasses that would be emitted from the process), also how the current social structures are needed to manage nuclear waste, ect. This is all very troubling and this is why my primary goal in life isn't to just oppose civilization or try to tell people to reject technology ect. I just wish that people would be more willing to criticize and reject the behaviors that people just take for granted, like police violence, familial repression, environmental destruction, work, capital, ect. I certainly cannot blame people for wanting to use social institutions to their advantage because I do and have done the same.
In the end, it's impossible for any individual to take on the burden of having to care for "the weak", you can only really care for yourself and those around you.