One could argue the pillars of society have it worse: there is no need for a savior or to change. Yet it is highly doubtful missions will spring up for this mostly doomed class...or if they did, anyone would take advantage of them.

I am an ex-con. I spent only three years in prison thanks to a suspended sentence and the benefit of good behavior. I was released to a residential treatment center for six months, then to a halfway house in a depressed and small community in the south of the state, where I knew no one. And that last part was a good thing, for everyone who knew me, knew me as a liar, cheat, and thief.
On the morning of my arrival, the director picked me up, brought me back to the house, then suggested I get settled in while she did a little shopping. During my time in prison I had renewed my faith in Christ and found a still voice in me. In that moment "the voice" said, "Go with her," so I did.
A little background: At this stage I am in my sixties, with no ID (a long story), no work history for over ten years, no marketable skills, and, last but not least, a convicted felon. This earned me high marks on the recividism chart.
Three blocks from the house, not in town an hour, I see a man struggling to get some equipment in a truck and ask the director to stop. "The voice" again. She knows the man; he is the station manager for a Christian TV station. I have been working there for almost four years now and my liife is gaining some semblance of decency and light.
Only the extraordinary, as you noted, provided me a chance. Yet it was not the government or even a church program; just the mercy of God and my receptivity, by grace, to open to His direction. 
What I have found in the church is people acting like Christians: this is what Jesus took issue with the Jews. They had not opened to a genuine love of neighbor but to the law of loving their neighbor in organized programs. It appears many Christian take the gifts of the spirit as behavior modification rules. This is external and without depth. There seems to be no notice or observation to actually dying to self, of letting go of attachments. Beliefs overpower surrender.