Yes. Have you seen the quality of NYC shelters? I have. I wouldn't stay in one if necessary. I'd rather try my luck outside as long as I don't freeze to death. Violence, SA, theft and robbery are daily occurrence, next to vermin and disease. Most of the money intended for it is usually redirected as high wages for the management (which tends to live well of their proceeds, while often the absolute minimum isn't even given to the shelters themselves. There is little to no oversight.

There is a tendency to stop looking into it, after one has received calming assurance. But the right response to "No need to worry, there is shelter for everyone" is to question the quality of those shelters. If most homeless tend to stay out of them, they don't seem work as intended. Having visited some work wise, I wouldn't stay in one, simply too dangerous.

So unless there is a broader social reform, there is little change to the current situation to be expected.