alt Hacker NewsThis is a fallacy because it's not the same people homeless each year. There's substantial turnover in the homeless population.
"Replace services with direct transfers" would eliminate homelessness for 1 year. Then new people would become homeless, and you'd have no services, because your budget is already committed to the "year 1 homeless-cohort."
The money spent grows quadratically (not linearly) over time.
If you only gave $100k to the people who are currently homeless that year, then (ignoring that people would start keeping themselves homeless for several years on purpose) you'd still end up better than the present situation.