A standard consequentialist model for welfare is that 1) each person cares about every person's well-being at least a little, but cares about themself much more and 2) a dollar goes farther at purchasing increased well-being in the hands of a poor person than in the hands of a well-off person.

For example, say that each person values themself 100x as much as they value another person and a dollar goes 10x farther in the hands of a poor person than in the hands of a well-off person. From the point of view of a typical well-off person, me getting a dollar is worth 100 utils (let's say, to set the scale), some other well-off person getting a dollar is worth 1 util (since I value them 1/100 as much), and some poor person getting a dollar is worth 10 utils (since I value them 1/100 as much and they get 10x as much well-being per dollar).

So one well-off person giving a dollar to a poor person is net negative according to the giver's preferences: -100 utils from losing a dollar plus 10 utils from a poor person getting a dollar (-100 + 10 = -90). But if 100 well-off people coordinate to each give a dollar to a different poor person then that is net positive or according to the preferences of one of those well-off people: -100 from losing a dollar, minus 99 from 99 other well-off people each losing a dollar, plus 1000 from 100 poor people each getting a dollar (-100 - 99 + 1000 = +801).

In this model, giving welfare is a coordination problem. From the point of view of one well-off person, it would be best to give nothing while all the other well-off people gave alms to the poor and worst to give to the poor while all the other well-off people gave nothing, and of the two coordinated solutions the one where everyone gives is better than the one where nobody gives. People used to try to solve this coordination problem by reputation, norms, etc. (judging almsgivers as virtuous and non-givers as vicious), but it gets harder to make that sort of solution work as society gets larger and more diffuse, with more interactions happening through impersonal markets and rule-based governments rather than within tightly-knit communities. Now people try to solve the coordination problem by getting the government to force everyone to participate.

The traditional forms of charity are church and family. The same goes for groups like unions but unions are usually not well regarded by the people who advocate for more charity.

Part of the idea of a secular nationstate is that the prime loyality of people isn't to their church community or their family but to the nationstate. We consider different loyalities to be a sign of corruption. The politician or judge who puts the interest of his church community or his family before the interested of the nationstate is a huge problem.

Lack of loyality to the nationstate is a huge reason why many African countries don't work. When government officials don't act in the interests of the nation but their clan the government doesn't work effectively.

Our secular nationstates do get loyality by engaging in actions like providing welfare for their citizens. Providing health care, both by regulating it and by organizing payment, is the same. In Hansian's terms it shows that the nation cares for your wellbeing and it gets loyality as return.

There are similar reasons why an organisations like Hamas or the Muslim brotherhood does a lot of charity. It's what you need to do for getting a certain kind of loyality.

In other words, welfare exists because destroying welfare is a coordination problem

That doesn't explain why the laws that create welfare were passed in the first place and why laws get sometimes passed to increase welfare payments.

Meta remark: I don't understand the thing where I post something on my personal blog, explicitely write that it is speculation and I want other people's insights, and get immediately downvoted. This is exactly the sort of thing I hated about the old LessWrong and that IMO contributed to its demise.

Theory 4: people only care about poor people in the in-group. People like welfare when it helps people similar to themselves, and hate it when it "enables" stupid lazy people in the out-group. This is often heavily racial - the countries with the most generous welfare systems are also the most racial homogeneous. So whether one votes in favor of welfare is determined by who they imagine being helped. This can often change rapidly, leaving public opinion on welfare to be a confusing mess.

All Western-style democracies have welfare systems, which means that presumably the majority is interested in having a welfare system, despite that it is a financial net negative for most people (that is, most people pay more money in taxes for welfare than receive welfare, I think?) So, apparently most people care about the poor.

The poor and the not-poor aren't completely distinct and disconnected groups. Few people are rich enough to be able to to absorb all the impacts of becoming unable to work, and many working people have non working relatives. State welfare can be seen as a way of insuring against the former, and spreading the cost of the latter. So there are selfish motivations, up to a point.

And it's typical of insurance schemes that most subsribers are nett losers -- that is hardly a bug. And the shear scale of state welfare, seen as insurance, is a rather desirable feature that is likely to go missing in the alternatives.

This theory is mostly true, but rather than being cynical about people caring about poor people, we should be cynical about the more general concept of people caring about stuff.

to the best of my knowledge

You need to improve on this. Also, consider the costs of welfare. And look into the history a bit more.

Many of the functions of welfare were provided by mutual societies, not by charities as we know them. You might find looking into who introduced welfare systems and why quite informative. It often had little to do with charitable notions and more to do with reducing crime.