I have a bit of experience with this. I worked in social services doing employment assistance for some time. The hard core long term homeless are not their base client. You're right... those people don't go to the places where it's offered.
The typical client I helped set up with voice mail is the temporary homeless surfing from couch to couch, or sleeping in their car, and even those who are unemployed and not yet homeless, but have nevertheless had their phone bills go unpaid too long and they were cut off. The phone bill is one of the first things to go when money runs short, but when it comes to finding a job, it's a necessity.
They're offering 350 voice mail boxes in Washington DC. I think there's probably that many takers.
I worked right next door to you, in Human Services. Many of these people have no dependents, so food stamps run out after three months. Oddly enough, most of them didn't gripe about that. Especially veterans.
The thought of veterans on the street makes me inexplicably sad. Whether it's by choice or not.
I hope you are wrong, that there aren't really 350 recent needs in DC. Logic tells me, though, that you're correct.