The problem is that unlike people with homes (who are generally stuck in one place — wherever their home is), homeless people have nothing holding them to places inhospitable to their presence. So the homeless diffuse around countries, drifting from place to place, until they settle in an "attractor" city — a local optimum for homeless hospitable-ness (with more social-welfare programs for them; less police suppression of homelessness; milder climate; more walkability; etc.)
These attractor cities then experience undue burden of having to fight basically an entire country's homeless problem using only municipal (rather than state or federal) tax dollars; where every social welfare program they add, creates "induced demand" by making them into more of an attractor city for surrounding homeless populations.
The homed people living in these attractor cities, experience a reality where they can choose to vote in progressive parties that throw however-much money at the problem of homelessness — adding public housing, shelters, food programs, jobs programs, etc. — with the only result being ever more homeless people showing up in the city. Which causes these homed people in these attractor cities to get burned out on the idea of solving homelessness.
(An aphorism: "the river sees more water than the lakebed.")
AFAICT, the only way out of the problem, is to
1. set aside federal funding for solving homelessness (which could be done easily enough in the US if anyone without a permanent residence in a particular state, were interpreted by federal law as not being "of" any state, but still a US national, and therefore a person the federal government has a duty to take care of directly — basically the same as how the federal government does insurance, postal delivery, etc. for actively-serving members of the military); and then
2. use that funding to start offering social-welfare programs for homeless people in places that don't already have large homeless populations, with the aim of creating more, smaller attractors, dispersing the homeless population, so that it's not all having to be dealt with by just the one or two natural attractor cities per country.
Yes, both of these things are big political lifts. I see it as something like the federal highway system, or for that matter, like military bases: cases where it's in all our best interests to relinquish local NIMBYesque control, in favor of top-down eminent-domain use of land for serving the needs of the country as a whole.