Most people who are mentally ill don't want to harm other people. The fact that they sometimes do, however, is indisputable. I would encourage you to do some research on homelessness in America and specifically on the chronic homeless. I've worked off-and-on with the homeless community for several years, and homeless people are not out to get anyone, they need help, it's just that providing that help is not nearly as simple as you make it out to be. Mental illness is so difficult to deal with partly because of its insidiousness, rather than what you are apparently imagining as "the truly deranged". Addiction is a big challenge as well, and the causative relationships are anything but simple and straightforward. Many people abuse substances to self-medicate for underlying mental health problems. Being homeless can create mental health problems as a consequences of the traumas associated with it.

Trying to wrap up complicated issues with a bow and dismiss them as simple is not a realistic or informed viewpoint. Do some research and then get back to me.