From what I've read, at the national level the incidence of serious mental illness in the homeless population are around 20 - 25%. So the numbers for San Francisco seem reasonable.
There's another option - we can not rely on self-reporting and just acknowledge that we don't know. It's not satisfying, but it sucks less than creating public policy based on bad/misleading information. I doubt very much we can fix 28% of homelessness by handing out free PO boxes.
It's literally like any other disease. We don't know how many people actually get the flu every year. We only know how many people self-report getting the flu, and how many people seek treatment. Self-reported surveys are used all the time in epidemiology and we manage to form effective public policy based on that.
So sure, there's definite bias when it comes to self-reporting, but your suggestion is to ignore what data we do have and form public policy based on what? How you feel? First principles?