"Housing" ... mostly a NIMBY problem. AKA not a small group of capitalists ruining it for everyone else, but a bunch of soccer moms who hate any densification and insist that their neighborhood must stay the same forever, weaponizing environmental laws to stop any development. Notably, cities that build nonetheless, be it Austin, TX, or Warsaw, Poland, buck this otherwise very widespread trend.
"Education, healthcare" ... depends on the country involved, plus the extent of healthcare you can now get is vastly bigger due to scientific progress. Forty years ago, HIV infection was a death sentence and most cancers too.
At any given time of year there are ~23 empty housing units for every single homeless person in the US, even close to 3 per homeless person in Los Angeles and New York City; somehow the constant, coordinated line of "supply constraints" and "grr NIMBYs!" doesn't hold up to the actual data
Vacant doesn't mean available. Apartments that are being shown to renters are vacant. Houses for sale are vacant. Houses that have sold or rented but haven't been moved into are not available.
It would be impossible to house homeless in vacant houses. Sellers and landlords would never accept it because would make it hard to find people. The short-term rentals would be hard to mange, not good for the homeless to always be moving, and easily go wrong. Is the government going to compensate for damages? For delayed moving into new house? It is almost certainly cheaper to build the homeless housing.
"Education, healthcare" ... depends on the country involved, plus the extent of healthcare you can now get is vastly bigger due to scientific progress. Forty years ago, HIV infection was a death sentence and most cancers too.