As a comparison: The F-35 fighter program is expected to cost $1.5 trillion over 55 years. That's $27 billion per year, or about three times as much as it would cost to fix homelessness.
Let's compare with what you're advocating, which appears to be the status quo. The existence of homeless people (not all of whom are drug addicts, by the way) shows that drug criminalisation doesn't work.
Homeless people don't need more suffering to finally get their act together. Life on the streets is plenty uncomfortable and dangerous, and nobody operating according to your simplistic model of sticks & carrots would chose it.
Further criminalising behaviour you don't like such as being badly dressed or prone to shouting random, schizophrenia-induced nonsense seems not just heartless, but rather close to dividing society into your kind and "subhumans" to be exterminated. Because if you don't think they deserve food or shelter, and you want to drive them out of cities (and, presumably, towns) where are they supposed to go? I want to be as charitable as possible here, but "push[ing] people out of the public sphere" seems to be a euphemism to bus them into the desert and not watch them die.
Being slightly annoyed by homeless people is also the least you can do if you actually believe in the "public sphere" as a community of citizens, and not just the place to get cheap takeout. If others' suffering cuts into your bliss, either through its annoying smells and sounds or because there are remnants of empathy in play, that should be motivation to solve the problem. There are many countries poorer than the US that have far smaller homeless populations, so it is entirely possible.