The situation of homeless is not as bad in countries with better systems as it is in the US, precisely because, as a tendency, their situation can be improved and less is done in the US than in other countries. The effectiveness of social welfare programs, health care, and so on, is a purely empirical question that can be measured effectively. Contrary to popular belief, these are not questions of political opinion. (What is a matter of political opinion, of course, is how much you want to spent and who is going to pay for it.)
You could e.g. start by helping the 70% of those homeless who, according to your own statistics, do not suffer from severe mental illness. That's already the vast majority.
As for the remaining 30%, the suggestion you're trying to make, that they are all hopeless cases, is outright bizarre. We know that schizophrenia can be treated with medicaments and that not all of the people diagnosed with schizophrenia refuse treatment. You could start by treating those who want to be treated, for example.
Finally, you still haven't bother to tell anyone what to do about the 30% you incorrectly portray as being untreatable, because they allegedly refuse treatment.
If your suggestion is to put them into mental institutions against their will, even when they pose no threat to the public or themselves, then my reply is that you need to check your moral compass. This is prohibited by law and by very basic moral considerations. If you have another suggestion, rather than stating the obvious truism that not every schizophrenic can fully adapt to society or be treated effectively, then I'm all ears, and so are myriads of medical professionals and street workers.
Right now, to me you merely come across as someone who doesn't really have anything to say about the topic. As I've said, it works better in many other countries, so your idea of not helping at all does not really seem to be very convincing to me.