My understanding of how it worked was that if you were functional enough and willing, you could walk into a certain building and they'd get you an apartment very quickly, although not sure if on the spot (I didn't work that part, just repairs part). I sometimes moved big bundles of keys for newly vacated or repaired apartments from the real estate building to the social workers in the homeless building they can then give out to new tenants.
I've now lived San Francisco for 5+ years and Helsinki basically does not have homeless people compared to what I've seen here.
I wondered a lot why California seems to be failing at the homeless problem. I see at least one comment here in threads that is saying that homeless are drug addicts and should be forced to go into rehab as a condition to give a home. While I was working for Helsinki I rarely heard anyone suggest the people being given homes needed to pay that back somehow, it was seen as obvious that the main problem is not having a home and the other problems can be dealt with later, and it's inhumane to make demands.
I don't know if "housing first" would actually work in California. Housing is super expensive, and I think California also has a lot more homeless than Finland ever did.
The ex-homeless tenants tended to need more repairs and care. I remember some funny/weird stories like we had a woman who could not use the toilet in her apartment because it was bright green and that caused her panic attacks. And some other tenant who painted literally everything (ceilings, windows, cabinets, furnite, floors, etc.) black.