The main problem is that new housing is largely banned as established wealth that already have detached homes vote for politicians that promise to not allow any new homes near them.
People could move to the midwest and live very cheaply if that were the issue.
The true cause of homelessness is mostly mental health issues, at least in these big cities.
Or do you think buying bus tickets for people is some sort of impossible to do thing?
Even if we were looking at this from an utterly cynical, purely financial viewpoint, the people paid to help the homeless are effectively property managers and they benefit from housing the homeless, not letting them sleep on the street.
A more modern example I recently stumbled across maybe 2 weeks ago was this crime story of this wicked nurse working in the ICU who was thrashing patients on his floor with air embolisms. The reason, the guy wanted to prolong the patients stay to ensure more overtime hours https://youtu.be/dgWcplHgAjU?t=645. Absolutely unbelievable.
You could hire people to investigate the problem and provide data. And then you could use that money to implement small scale (locality level) experiments to see what works. And still have billions left over.
For some reason, the whole concept of "Do more of what works, and less of what doesn't" seems lost on the people involved.