Here's a report on homelessness and its relationships to severe mental illness
http://www.nationalhomeless.org/factsheets/Mental_Illness.pdf
Of course, the report is arguing in favor of more statist funding, but I think it actually proves an opposing point.
While no one wants to see homeless mentally ill people wandering around the streets, the facts are this is the situation we have right now with a massive state.
Do we currently see dead homeless mentally ill bums lining the streets?
No, we do not.
Even severely mentally ill people have the capability to survive, even if it is in a poor condition.
Since we know that being homeless and mentally ill does not equate to a death sentence, and since we know that the State doesn't care for these people right now, we can say that the severely mentally ill that are unable to house themselves would CLEARLY be in a better position in a free market state of affairs.
Since we know private charitable donations would be substantial in a free market, we can say with certainty that private charities would be in a far better position to take care of these people that the State is currently neglecting. Further, panhandlers would be in a superior position as well. If a mentally ill homeless person is panhandling for his survival, its clear that if the general public was wealthier in general, the panhandler will increase the donations he receives.
Also, we can't forget the family unit. Mentally ill people were born into the world so therefore it stands to reason they typically have some family that can offer them some level of support.
In old age or under such conditions as a person truly becomes incapable of caring for themselves in any respect, private charities and church organizations would be there to pick up the slack. Further, no one is going to walk past a 70 old starving homeless grandma lying on the road and leave her for dead. I have enough faith in humanity that someone will step forward to make sure she gets put in a charity home or even take care of her themselves.
Even today with our outrageous criminal tax burden, we have privately run charitable homes for those that are totally destitute in old age. As society grows richer, the problem of caring for the mentally ill would remedy itself. Families would ultimately become wealthy enough to establish private trusts to care for their indigent members into old age subsidized by public charitable contributions.
American history is clear that in the absence of State programs, such people will survive.
I would also look to the church model. Here we have massive castles of religious worship funded entirely through charitable donations in nearly every town in America. It stands to reason that if a society can fund these castles and pastors voluntarily through charity, they can take care of the sick and indigent as well.