Actually, only 35% of San Francisco's homeless population has some sort of Psychiatric condition. According to a survey of SF's homeless the biggest barriers to finding work (as of 2015) were:
28% said no permanent address
20% said alcohol or drug use
17% said disability
14% said age
13% said need for clothing/shower facilities
...
9% said mental health concerns
So it does seems like lack of housing is the largest barrier for SF's homeless population. Additionally, mental illness is most prevalent amongst the chronically homeless, which only make up about 25% of the homeless population in SF. The vast majority of homeless people are homeless for months not years, and the many programs to aid them in finding housing and preventing homelessness in the first place have a significant measurable effect on the average number of homeless shelter entries and length of stay.
Source: https://sfgov.org/lhcb/sites/default/files/2015%20San%20Fran...