In our state, it's easier to find a primary care provider for a Medicaid pt than a Medicare fee for service pt. The case workers assisting with the mentally ill and those with qualifying chronic conditions are pretty akamai to set their patients up with the more integrated Medicaid/Medicare providers such as Kaiser Permanente and HMSA (Blue Cross Blue Shield). Finding a PCP for such patients will be no different from finding a PCP if you had straight forward HMO coverage.
More recently, the homeless population case workers in the community or hospitals are following suit: when a client is auto assigned by the state to a random Medicaid provider (usually a small company with no healthcare system behind their name such as Aloha Care or Ohana) they assist and encourage the pts to have their coverage transferred to providers such as Kaiser or HMSA.
In addition, most of these ginormous insurances have built teams and programs to minimize the accessibility issue, hospitalization, re-admission...etc. Obviously for reimbursement issues but mostly because it ultimately decreases healthcare costs in the long run: preventative health is key. So they bombard the "at risk" population with community outreach, SWs and RNs go to the field, they provide notices, appt reminders, and even offer free transportation to all medical appointments.
I apologize, I am not trying to be offensive and I am not saying homeless people are not underserved, I am CERTAIN some are... I am just replying to the "homeless people should qualify for free healthcare" idea. I strongly believe that would be better assessed case by case, at least here in our state.