Absolutely. Twice a year my church (as well as other churches around Richmond) hosts homeless families who are 1) working and 2) seeking a place to live. The majority are single mothers, and of them most are minorities. They are not alcoholics, drug users or mentally ill. While yes, some are, this group is screened pretty carefully (because for a week, they actually sleep in the church building).
Many homeless are children. And sure in some cases they are the children of the alcoholics, drug users and minorities. But the children can hardly be blamed for that.
The problems you list are very real, very prevalent.
But they do not address every "homeless" situation.
There would be NO government support of any kind. That way, religious people would have a real live cause to work on and the "homeless" would be dispersed and not gather at bus stations, skid row, etc. The homeless could repay the church/temple by doing odd jobs, mowing the lawn, washing windows, litter pick-up, etc, while getting training for jobs, getting detoxified and prepared to enter the mainstream.
Those who chose to opt out of the program would be on their own to survive as best they could, knowing that the option of joining the program was always there.