That is such a common pattern. Growing up, I saw the same happen with the German welfare system: there's enough of a social safety net in that country that no one would have to be homeless. But, to access the social safety net, you have to navigate some minimal amount of bureaucracy and paperwork.
That bar isn't very high, but if you have at least that minimal skill with paperwork literacy, you are also much more likely to have a job and much less likely to need the welfare.
So all in all, Germany still has homeless people, despite welfare programs generous enough in principle that no one needs to be homeless.
Addendum: Singapore has an ingenious system to ration the amount of subsidies you get on hospital bills without any bureaucracy. The basic idea is that when you go for a hospital stay, you get to pick how much etxra creature comforts you are getting, like your own private room vs an open word with many beds. The copay for the fancier options rises 'progressively' enough, that the absolute amount of government subsidy goes down. People tend to self-sort voluntarily.
The medical care you receive is the same for any of the options.
I can imagine an argument that this is somehow against the dignity of poor people to put them into comparatively crowded wards, and that everyone should get the same treatment (like eg the British NHS does, modulo the option to go to a private hospital for enough money). But I suspect many of the poorer people are happier to put up with that 'indignity' than with paperwork and privacy-invading bureaucratic means testing.
In total, adding up private and public expenditures, Singapore spends about half as much on healthcare as a proportion of GDP than the UK does, which spends about half as much as the US. Medical outcomes are no worse in the cheaper systems.