There are two major homeless shelters in my area I've volunteered for. One does exactly what you prescribe and tries to separate out those who can be easily put on a track to success from those who need some food and a place to crash. The other seems to lump everyone in together and focus on meeting basic needs, although that may not be entirely fair to their operation. That one is closer to downtown, not that far from the football stadium, and I think they are a bit overwhelmed by numbers, which are small compared to West Coast numbers.
The womens' shelters seem to have a higher percentage of people just in temporary bad straits due to domestic violence. But they have plenty of crazy folks and addicts too.
A decade ago downtown was in rough shape simply because of the amount of beggars, plus the sordid Occupy camp. The FBI eventually arrested several of the Occupy leaders here for conspiracy to blow up one of the I-75 bridges. And after Cleveland legalized gambling the city has tried to clear out public square of vagrants more aggressively due to pressure from the casinos. It seemed to work well at first, and downtown became a pleasant place to visit again. Well, relatively.