> I think that most homeless people have severe mental illness

This is another stigma that's counter-productive. The real number is somewhere between 20-25% suffering from severe mental illness, which is still very high, though not so high when you consider 22% of Americans suffer from some kind of a mental illness (only 6% of Americans are severely mentally ill). Mental illness is only the third largest cause of homelessness for single adults.

To add to your point: poverty causes something like a 13% drop in the IQ, similar to pulling an all-nighter every day for a week.