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.
Thus, cutting all budget for cash transfers could increase homelessness by 80%! (Are those 14,498 now-homeless people eligible for this year's $100k? They can't be, since you already spent it on the 7,973 currently-homeless people).
Model the situation as "X(t) people become homeless at year t, Y(t) people become housed at time t," and you'll see the most important metric is "how much can we decrease X and increase Y per dollar spent"?
"Number of dollars per current homeless" is not really meaningful at all.
It's unlikely there's a trivial arbitrage to eliminate homelessness in a city full of engineers.
It’s like history was the first casualty of late stage capitalism and no one knows about the new deal or the war on poverty.
Don't you need to spend on bureaucracy to figure this out? That then increases the cost.