What matters is that people are voting with their wallets.
People need a solution to rampant crime. Take for example SA, where private security forces are widespread.
The degree of deleriction of duty by public servants has been tremendous in the last 2 years. The NYC Major recently was on the record stating that "there is no security problem in NYC" meanwhile official City hall communication channels have implemented a buddy system where city hall employees can home in groups [1]
If public servants can't provide security for people, there should not be a surprise that the private servicers would emerge.
[1]https://www.nbcnewyork.com/news/local/nyc-city-hall-employee...
I don't think we want to use as a model a place where police services are only available to the wealthy.
Someone who is looking at the full situation would realize that hiring private security does not cause the homeless population to suddenly evaporate. An outsider might start to get the impression that the fundamental right of the upper-middle class is the right not to be reminded that people in poverty exist.
In addition, it's not like these people don't have sympathy for the plight of the poor: they voted to spend vast sums on tax money trying to fix the root cause of the problem. It's not their fault that these efforts are failing.
You're looking at the most obvious symptom - people with no home - and saying "well if we just treated that symptom with all this money there would be no problem." But the root cause is an order of magnitude larger than that.